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(Sorry, I fell short of imagination in the title of the question)

One more time I am surprise by what I do not understand. Here is a working code

def one_dimension_less(P,c):
    A,b = P
    m = A.nrows(); 
    M = range(0,m)
    n = A.ncols() 
    N = [i for i in M if A[i,:]*c < 0]
    Z = [i for i in M if A[i,:]*c == 0]
    P = [i for i in M if A[i,:]*c > 0]
    p = Z + [(i,j) for i in N for j in P]
    r = len(p)
    D = Matrix(r,n); d = Matrix(r,1)
    for i in range(0,r):
        if not isinstance(p[i],tuple):
            D[i,:] = A[p[i],:]
            d[i] = b[p[i]]
        else:
            (s,t) = p[i]
            D[i,:] = (A[t,:]*c)*A[s,:] - (A[s,:]*c)*A[t,:]
            d[i] = (A[t,:]*c)*b[s] - (A[s,:]*c)*b[t]
    return (D,d)

with a good call :

A=matrix([[1,2,3],[1,2,1],[2,2,1]])
b=vector([1,1,1])
c=vector([1,0,0])
one_dimension_less(P,c)

Until there it's a simple projection on c. As, in the exemple, c could be (1,0,0), (0,1,0) or (0,0,1) it comes to my mind that I can prepare the call to the function according to :

A=matrix([[1,2,3],[1,2,1],[2,2,1]])
b=vector([1,1,1])
c=zero_vector(3)
c[1]=1
show(c)
one_dimension_less(P,c)

So in changing the index of 1 in c, I can change the projection. Now It cames to my mind that in passing only the index in the function I can obtain directly what I expect without more preparation than the A matrix and the c vector. For that I change the function to

def one_dimension_less(P,param):
    A,b = P
    m = A.nrows(); 
    M = range(0,m)
    n = A.ncols() 
    c=zero_vector(m)
    c[param]=1
    N = [i for i in M if A[i,:]*c < 0]
    Z = [i for i in M if A[i,:]*c == 0]
    P = [i for i in M if A[i,:]*c > 0]
    p = Z + [(i,j) for i in N for j in P]
    r = len(p)
    D = Matrix(r,n); d = Matrix(r,1)
    for i in range(0,r):
        if not isinstance(p[i],tuple):
            D[i,:] = A[p[i],:]
            d[i] = b[p[i]]
        else:
            (s,t) = p[i]
            D[i,:] = (A[t,:]*c)*A[s,:] - (A[s,:]*c)*A[t,:]
            d[i] = (A[t,:]*c)*b[s] - (A[s,:]*c)*b[t]
    return (D,d)

Now the call

one_dimension_less(P,0)

return the error :

unsupported operand parent(s) for *: 'Full MatrixSpace of 1 by 3 dense matrices over Integer Ring' and 'Ambient free module of rank 6 over the principal ideal domain Integer Ring'.

I think to understand that this is a ring problem but I do not see why it works from outside and not from inside the function.