let M1, M2 be Matrix of n,K; :: thesis: ( ( for i, j being Nat st [i,j] in Indices A holds
M1 * i,j = - (A * i,j) ) & ( for i, j being Nat st [i,j] in Indices A holds
M2 * i,j = - (A * i,j) ) implies M1 = M2 )
assume that
A14: for i, j being Nat st [i,j] in Indices A holds
M1 * i,j = - (A * i,j) and
A15: for i, j being Nat st [i,j] in Indices A holds
M2 * i,j = - (A * i,j) ; :: thesis: M1 = M2
( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by Th25;
then A16: ( Indices A = Indices M1 & Indices M1 = Indices M2 ) by Th25;

now

let i, j be Nat; :: thesis: ( [i,j] in Indices A implies M1 * i,j = M2 * i,j )
assume [i,j] in Indices A ; :: thesis: M1 * i,j = M2 * i,j
then ( M1 * i,j = - (A * i,j) & M2 * i,j = - (A * i,j) ) by A14, A15;
hence M1 * i,j = M2 * i,j ; :: thesis: verum

end;

hence M1 = M2 by A16, Th28; :: thesis: verum