let a be complex number ; :: thesis: for M being Matrix of COMPLEX holds (a * M) *' = (a *' ) * (M *' )
let M be Matrix of COMPLEX ; :: thesis: (a * M) *' = (a *' ) * (M *' )
reconsider aa = a as Element of COMPLEX by XCMPLX_0:def 2;
A1: len (a * M) = len M by Th3;
A2: len ((a * M) *' ) = len (a * M) by Def1;
len ((a *' ) * (M *' )) = len (M *' ) by Th3;
then len ((a *' ) * (M *' )) = len M by Def1;
then A3: len ((a * M) *' ) = len ((a *' ) * (M *' )) by A2, Th3;
A4: len M = len (M *' ) by Def1;
A5: width (a * M) = width M by Th3;
A6: width ((a * M) *' ) = width (a * M) by Def1;
width ((a *' ) * (M *' )) = width (M *' ) by Th3;
then width ((a *' ) * (M *' )) = width M by Def1;
then A7: width ((a * M) *' ) = width ((a *' ) * (M *' )) by A6, Th3;
A8: width M = width (M *' ) by Def1;

now

let i, j be Nat; :: thesis: ( [i,j] in Indices ((a * M) *' ) implies ((a * M) *' ) * i,j = ((a *' ) * (M *' )) * i,j )
assume [i,j] in Indices ((a * M) *' ) ; :: thesis: ((a * M) *' ) * i,j = ((a *' ) * (M *' )) * i,j
then A9: ( 1 <= i & i <= len (a * M) & 1 <= j & j <= width (a * M) ) by A2, A6, Th1;
then A10: [i,j] in Indices (a * M) by Th1;
A11: [i,j] in Indices M by A1, A5, A9, Th1;
A12: [i,j] in Indices (M *' ) by A1, A4, A5, A8, A9, Th1;
((a * M) *' ) * i,j = ((a * M) * i,j) *' by A10, Def1;
hence ((a * M) *' ) * i,j = (aa * (M * i,j)) *' by A11, Th4
.= (aa *' ) * ((M * i,j) *' ) by COMPLEX1:121
.= (a *' ) * ((M *' ) * i,j) by A11, Def1
.= ((a *' ) * (M *' )) * i,j by A12, Th4 ;
:: thesis: verum

end;

hence (a * M) *' = (a *' ) * (M *' ) by A3, A7, MATRIX_1:21; :: thesis: verum