thus ( len pD = len M implies ex M1 being Matrix of n,m,D st
( 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 ) ) ) ) ) :: thesis: ( not len pD = len M implies ex b₁ being Matrix of n,m,D st b₁ = M )

proof

reconsider M9 = M as Matrix of len M, width M,D by MATRIX_2:7;
reconsider V = n, U = m as Element of NAT by ORDINAL1:def 13;
defpred S₁[ set , set , set ] means for i, j being Nat st i = $1 & j = $2 holds
( ( j <> c implies $3 = M * (i,j) ) & ( j = c implies $3 = pD . i ) );
assume A1: len pD = len M ; :: thesis: ex M1 being Matrix of n,m,D st
( 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 ) ) ) )
A2: for i, j being Nat st [i,j] in [:(Seg V),(Seg U):] holds
ex x being Element of D st S₁[i,j,x]

proof

let i, j be Nat; :: thesis: ( [i,j] in [:(Seg V),(Seg U):] implies ex x being Element of D st S₁[i,j,x] )
assume A3: [i,j] in [:(Seg V),(Seg U):] ; :: thesis: ex x being Element of D st S₁[i,j,x]

now

per cases ( j = c or j <> c ) ;

case A4: j = c ; :: thesis: ex x being Element of D st S₁[i,j,x]

A5: rng pD c= D by FINSEQ_1:def 4;
len M = n by MATRIX_1:def 3;
then i in Seg (len pD) by A1, A3, ZFMISC_1:106;
then i in dom pD by FINSEQ_1:def 3;
then A6: pD . i in rng pD by FUNCT_1:def 5;
S₁[i,j,pD . i] by A4;
hence ex x being Element of D st S₁[i,j,x] by A6, A5; :: thesis: verum

end;

case j <> c ; :: thesis: ex x being Element of D st S₁[i,j,x]

then S₁[i,j,M * (i,j)] ;
hence ex x being Element of D st S₁[i,j,x] ; :: thesis: verum

end;

end;

end;

hence ex x being Element of D st S₁[i,j,x] ; :: thesis: verum

end;

A7: for i, j being Nat st [i,j] in [:(Seg V),(Seg U):] holds
for x1, x2 being Element of D st S₁[i,j,x1] & S₁[i,j,x2] holds
x1 = x2

proof

let i, j be Nat; :: thesis: ( [i,j] in [:(Seg V),(Seg U):] implies for x1, x2 being Element of D st S₁[i,j,x1] & S₁[i,j,x2] holds
x1 = x2 )
assume [i,j] in [:(Seg V),(Seg U):] ; :: thesis: for x1, x2 being Element of D st S₁[i,j,x1] & S₁[i,j,x2] holds
x1 = x2
let x1, x2 be Element of D; :: thesis: ( S₁[i,j,x1] & S₁[i,j,x2] implies x1 = x2 )
assume that
A8: S₁[i,j,x1] and
A9: S₁[i,j,x2] ; :: thesis: x1 = x2
reconsider i = i, j = j as Element of NAT by ORDINAL1:def 13;
A10: ( j = c implies x1 = pD . i ) by A8;
( j <> c implies x1 = M * (i,j) ) by A8;
hence x1 = x2 by A9, A10; :: thesis: verum

end;

consider M1 being Matrix of V,U,D such that
A11: for i, j being Nat st [i,j] in Indices M1 holds
S₁[i,j,M1 * (i,j)] from MATRIX_1:sch 2(A7, A2);
reconsider M1 = M1 as Matrix of n,m,D ;
take M1 ; :: thesis: ( 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 ) ) ) )

A12: now

per cases ( n = 0 or n > 0 ) ;

suppose A13: n = 0 ; :: thesis: ( len M = len M1 & width M1 = width M )

then len M1 = 0 by MATRIX_1:def 3;
then A14: width M1 = 0 by MATRIX_1:def 4;
len M = 0 by A13, MATRIX_1:def 3;
hence ( len M = len M1 & width M1 = width M ) by A13, A14, MATRIX_1:def 3, MATRIX_1:def 4; :: thesis: verum

end;

suppose A15: n > 0 ; :: thesis: ( len M = len M1 & width M = width M1 )

then A16: width M = m by MATRIX_1:24;
len M = n by A15, MATRIX_1:24;
hence ( len M = len M1 & width M = width M1 ) by A15, A16, MATRIX_1:24; :: thesis: verum

end;

end;

end;

Indices M9 = Indices M1 by MATRIX_1:27;
hence ( 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 ) ) ) ) by A11, A12; :: thesis: verum

end;

thus ( not len pD = len M implies ex b₁ being Matrix of n,m,D st b₁ = M ) ; :: thesis: verum