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) + (B * i,j) ) & ( for i, j being Nat st [i,j] in Indices A holds
M2 * i,j = (A * i,j) + (B * i,j) ) implies M1 = M2 )
assume that
A23: for i, j being Nat st [i,j] in Indices A holds
M1 * i,j = (A * i,j) + (B * i,j) and
A24: for i, j being Nat st [i,j] in Indices A holds
M2 * i,j = (A * i,j) + (B * i,j) ; :: thesis: M1 = M2

A25: now

let i, j be Nat; :: thesis: ( [i,j] in Indices A implies M1 * i,j = M2 * i,j )
assume A26: [i,j] in Indices A ; :: thesis: M1 * i,j = M2 * i,j
then M1 * i,j = (A * i,j) + (B * i,j) by A23;
hence M1 * i,j = M2 * i,j by A24, A26; :: thesis: verum

end;

Indices M1 = [:(Seg n),(Seg n):] by Th25;
then Indices A = Indices M1 by Th25;
hence M1 = M2 by A25, Th28; :: thesis: verum