let n be Nat; for K being Field
for M1, M2 being Matrix of n,K st M1 is_congruent_Matrix_of M2 & n > 0 holds
(M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2
let K be Field; for M1, M2 being Matrix of n,K st M1 is_congruent_Matrix_of M2 & n > 0 holds
(M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2
let M1, M2 be Matrix of n,K; ( M1 is_congruent_Matrix_of M2 & n > 0 implies (M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2 )
assume that
A1:
M1 is_congruent_Matrix_of M2
and
A2:
n > 0
; (M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2
consider M4 being Matrix of n,K such that
A3:
M4 is invertible
and
A4:
M1 = ((M4 @) * M2) * M4
by A1;
A5:
( len M4 = n & len ((M4 @) * M2) = n )
by MATRIX_0:24;
A6:
width ((M4 @) * M2) = n
by MATRIX_0:24;
A7:
( len M2 = n & width M2 = n )
by MATRIX_0:24;
A8:
( len (M4 @) = n & width (M4 @) = n )
by MATRIX_0:24;
then A9: ((M4 @) * (M2 + M2)) * M4 =
(((M4 @) * M2) + ((M4 @) * M2)) * M4
by A2, A7, MATRIX_4:62
.=
M1 + M1
by A2, A4, A5, A6, MATRIX_4:63
;
take
M4
; MATRIX_8:def 6 ( M4 is invertible & (M1 + M1) + M1 = ((M4 @) * ((M2 + M2) + M2)) * M4 )
A10:
( len ((M4 @) * (M2 + M2)) = n & width ((M4 @) * (M2 + M2)) = n )
by MATRIX_0:24;
( len (M2 + M2) = n & width (M2 + M2) = n )
by MATRIX_0:24;
then ((M4 @) * ((M2 + M2) + M2)) * M4 =
(((M4 @) * (M2 + M2)) + ((M4 @) * M2)) * M4
by A2, A7, A8, MATRIX_4:62
.=
(M1 + M1) + M1
by A2, A4, A5, A6, A10, A9, MATRIX_4:63
;
hence
( M4 is invertible & (M1 + M1) + M1 = ((M4 @) * ((M2 + M2) + M2)) * M4 )
by A3; verum