let K be Ring; :: 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 that
A1: len M1 = len M2 and
A2: width M1 = width M2 ; :: thesis: M1 - (M1 - M2) = M2
A3: ( len (- M1) = len M1 & width (- M1) = width M1 ) by MATRIX_3:def 2;
A4: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;

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

suppose A5: len M1 > 0 ; :: thesis: M1 - (M1 - M2) = M2

A6: len (0. (K,(len M1),(width M1))) = len M1 by MATRIX_0:def 2;
then A7: width (0. (K,(len M1),(width M1))) = width M1 by A5, MATRIX_0:20;
A8: M2 is Matrix of len M1, width M1,K by A1, A2, A5, MATRIX_0:20;
A9: M1 is Matrix of len M1, width M1,K by A5, MATRIX_0:20;
M1 - (M1 - M2) = M1 + ((- M1) + (- (- M2))) by A1, A2, A4, Th12
.= M1 + ((- M1) + M2) by Th1
.= (M1 + (- M1)) + M2 by A3, MATRIX_3:3
.= (0. (K,(len M1),(width M1))) + M2 by A9, MATRIX_3:5
.= M2 + (0. (K,(len M1),(width M1))) by A1, A2, A6, A7, MATRIX_3:2
.= M2 by A8, MATRIX_3:4 ;
hence M1 - (M1 - M2) = M2 ; :: thesis: verum

end;

suppose A10: len M1 = 0 ; :: thesis: M1 - (M1 - M2) = M2

then len (M1 - (M1 - M2)) = 0 by MATRIX_3:def 3;
hence M1 - (M1 - M2) = M2 by A1, A10, CARD_2:64; :: thesis: verum

end;

end;