let M1, M2 be Matrix of n,m,D; :: thesis: ( ( len pD = len M & len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( j <> c implies M1 * (i,j) = M * (i,j) ) & ( j = c implies M1 * (i,c) = pD . i ) ) ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( j <> c implies M2 * (i,j) = M * (i,j) ) & ( j = c implies M2 * (i,c) = pD . i ) ) ) implies M1 = M2 ) & ( not len pD = len M & M1 = M & M2 = M implies M1 = M2 ) )

thus ( len pD = len M & len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( j <> c implies M1 * (i,j) = M * (i,j) ) & ( j = c implies M1 * (i,c) = pD . i ) ) ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( j <> c implies M2 * (i,j) = M * (i,j) ) & ( j = c implies M2 * (i,c) = pD . i ) ) ) implies M1 = M2 ) :: thesis: ( not len pD = len M & M1 = M & M2 = M implies M1 = M2 )
proof
assume len pD = len M ; :: thesis: ( not len M1 = len M or not width M1 = width M or ex i, j being Nat st
( [i,j] in Indices M & not ( ( j <> c implies M1 * (i,j) = M * (i,j) ) & ( j = c implies M1 * (i,c) = pD . i ) ) ) or not len M2 = len M or not width M2 = width M or ex i, j being Nat st
( [i,j] in Indices M & not ( ( j <> c implies M2 * (i,j) = M * (i,j) ) & ( j = c implies M2 * (i,c) = pD . i ) ) ) or M1 = M2 )

assume that
A13: len M1 = len M and
A14: width M1 = width M and
A15: for i, j being Nat st [i,j] in Indices M holds
( ( j <> c implies M1 * (i,j) = M * (i,j) ) & ( j = c implies M1 * (i,c) = pD . i ) ) ; :: thesis: ( not len M2 = len M or not width M2 = width M or ex i, j being Nat st
( [i,j] in Indices M & not ( ( j <> c implies M2 * (i,j) = M * (i,j) ) & ( j = c implies M2 * (i,c) = pD . i ) ) ) or M1 = M2 )

assume that
len M2 = len M and
width M2 = width M and
A16: for i, j being Nat st [i,j] in Indices M holds
( ( j <> c implies M2 * (i,j) = M * (i,j) ) & ( j = c implies M2 * (i,c) = pD . i ) ) ; :: thesis: M1 = M2
for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j)
proof
let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
then A17: [i,j] in Indices M by ;
reconsider I = i, J = j as Element of NAT by ORDINAL1:def 12;
A18: ( J = c implies M1 * (I,c) = pD . I ) by ;
A19: ( J <> c implies M2 * (I,J) = M * (I,J) ) by ;
( J <> c implies M1 * (I,J) = M * (I,J) ) by ;
hence M1 * (i,j) = M2 * (i,j) by A16, A17, A18, A19; :: thesis: verum
end;
hence M1 = M2 by MATRIX_0:27; :: thesis: verum
end;
thus ( not len pD = len M & M1 = M & M2 = M implies M1 = M2 ) ; :: thesis: verum