let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds

M1 = M1 + (M2 - M2)

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies M1 = M1 + (M2 - M2) )

assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: M1 = M1 + (M2 - M2)

M1 = M1 + (M2 - M2)

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies M1 = M1 + (M2 - M2) )

assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: M1 = M1 + (M2 - M2)

per cases
( len M1 > 0 or len M1 = 0 )
by NAT_1:3;

end;

suppose A2:
len M1 > 0
; :: thesis: M1 = M1 + (M2 - M2)

then A3:
M1 is Matrix of len M1, width M1,K
by MATRIX_0:20;

M2 is Matrix of len M1, width M1,K by A1, A2, MATRIX_0:20;

hence M1 + (M2 - M2) = M1 + (0. (K,(len M1),(width M1))) by MATRIX_3:5

.= M1 by A3, MATRIX_3:4 ;

:: thesis: verum

end;M2 is Matrix of len M1, width M1,K by A1, A2, MATRIX_0:20;

hence M1 + (M2 - M2) = M1 + (0. (K,(len M1),(width M1))) by MATRIX_3:5

.= M1 by A3, MATRIX_3:4 ;

:: thesis: verum