(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.