let a be Element of REAL ; :: thesis: for n being Nat
for M being Matrix of n, REAL holds |:(a * M):| = (abs a) * |:M:|
let n be Nat; :: thesis: for M being Matrix of n, REAL holds |:(a * M):| = (abs a) * |:M:|
let M be Matrix of n, REAL ; :: thesis: |:(a * M):| = (abs a) * |:M:|
A1: ( len (a * M) = len M & width (a * M) = width M ) by MATRIXR1:27;
len ((abs a) * |:M:|) = len |:M:| by MATRIXR1:27;
then A2: len ((abs a) * |:M:|) = len M by Def7;
A3: for i, j being Nat st [i,j] in Indices |:(a * M):| holds
|:(a * M):| * (i,j) = ((abs a) * |:M:|) * (i,j)

proof

A4: Indices (a * M) = Indices M by MATRIXR1:28;
A5: Indices (a * M) = Indices M by MATRIXR1:28;
let i, j be Nat; :: thesis: ( [i,j] in Indices |:(a * M):| implies |:(a * M):| * (i,j) = ((abs a) * |:M:|) * (i,j) )
assume A6: [i,j] in Indices |:(a * M):| ; :: thesis: |:(a * M):| * (i,j) = ((abs a) * |:M:|) * (i,j)
A7: Indices |:(a * M):| = Indices (a * M) by Th5;
then A8: |:(a * M):| * (i,j) = abs ((a * M) * (i,j)) by A6, Def7
.= abs (a * (M * (i,j))) by A6, A7, A4, Th4
.= (abs a) * (abs (M * (i,j))) by COMPLEX1:65 ;
Indices |:M:| = Indices M by Th5;
then ((abs a) * |:M:|) * (i,j) = (abs a) * (|:M:| * (i,j)) by A6, A7, A5, Th4
.= |:(a * M):| * (i,j) by A6, A7, A8, A5, Def7 ;
hence |:(a * M):| * (i,j) = ((abs a) * |:M:|) * (i,j) ; :: thesis: verum

end;

width ((abs a) * |:M:|) = width |:M:| by MATRIXR1:27;
then A9: width ((abs a) * |:M:|) = width M by Def7;
( len |:(a * M):| = len (a * M) & width |:(a * M):| = width (a * M) ) by Def7;
hence |:(a * M):| = (abs a) * |:M:| by A1, A2, A9, A3, MATRIX_1:21; :: thesis: verum