let M be Matrix of COMPLEX ; :: thesis: (- 1) * M = - M
A1: len (- M) = len M by Th9;
A2: len ((- 1) * M) = len M by Th3;
A3: width (- M) = width M by Th9;
A4: width ((- 1) * M) = width M by Th3;

now

let i, j be Nat; :: thesis: ( [i,j] in Indices ((- 1) * M) implies ((- 1) * M) * i,j = (- M) * i,j )
assume [i,j] in Indices ((- 1) * M) ; :: thesis: ((- 1) * M) * i,j = (- M) * i,j
then ( 1 <= i & i <= len M & 1 <= j & j <= width M ) by A2, A4, Th1;
then A5: [i,j] in Indices M by Th1;
hence ((- 1) * M) * i,j = (- 1) * (M * i,j) by Th4
.= - (M * i,j)
.= (- M) * i,j by A1, A3, A5, Th10 ;
:: thesis: verum

end;

hence (- 1) * M = - M by A1, A2, A3, A4, MATRIX_1:21; :: thesis: verum