let S1, S2 be Matrix of n,m,D; :: thesis: ( ( for i, j being Nat st [i,j] in Indices S1 holds
S1 * (i,j) = M * ((nt . i),(mt . j)) ) & ( for i, j being Nat st [i,j] in Indices S2 holds
S2 * (i,j) = M * ((nt . i),(mt . j)) ) implies S1 = S2 )
assume that
A2: for i, j being Nat st [i,j] in Indices S1 holds
S1 * (i,j) = M * ((nt . i),(mt . j)) and
A3: for i, j being Nat st [i,j] in Indices S2 holds
S2 * (i,j) = M * ((nt . i),(mt . j)) ; :: thesis: S1 = S2

now :: thesis: for i, j being Nat st [i,j] in Indices S1 holds
S1 * (i,j) = S2 * (i,j)

let i, j be Nat; :: thesis: ( [i,j] in Indices S1 implies S1 * (i,j) = S2 * (i,j) )
assume A4: [i,j] in Indices S1 ; :: thesis: S1 * (i,j) = S2 * (i,j)
A5: [i,j] in Indices S2 by A4, MATRIX_0:26;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def 12;
S1 * (i,j) = M * ((nt . i9),(mt . j9)) by A2, A4;
hence S1 * (i,j) = S2 * (i,j) by A3, A5; :: thesis: verum

end;

hence S1 = S2 by MATRIX_0:27; :: thesis: verum