let K be Ring; :: thesis: for M1, M2, M3 being Matrix of K st len M1 = len M2 & len M2 = len M3 & width M1 = width M2 & width M2 = width M3 holds
M1 + M3 = (M1 + M2) - (M2 - M3)
let M1, M2, M3 be Matrix of K; :: thesis: ( len M1 = len M2 & len M2 = len M3 & width M1 = width M2 & width M2 = width M3 implies M1 + M3 = (M1 + M2) - (M2 - M3) )
assume that
A1: len M1 = len M2 and
A2: len M2 = len M3 and
A3: width M1 = width M2 and
A4: width M2 = width M3 ; :: thesis: M1 + M3 = (M1 + M2) - (M2 - M3)

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

suppose A5: len M1 > 0 ; :: thesis: M1 + M3 = (M1 + M2) - (M2 - M3)

then A6: M2 is Matrix of len M1, width M1,K by A1, A3, MATRIX_0:20;
A7: ( len (- M2) = len M1 & width (- M2) = width M1 ) by A1, A3, MATRIX_3:def 2;
A8: ( len (- M3) = len M1 & width (- M3) = width M1 ) by A1, A2, A3, A4, MATRIX_3:def 2;
A9: ( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 ) by MATRIX_3:def 3;
M1 is Matrix of len M1, width M1,K by A5, MATRIX_0:20;
hence M1 + M3 = (M1 + (0. (K,(len M1),(width M1)))) + M3 by MATRIX_3:4
.= (M1 + (M2 + (- M2))) + M3 by A6, MATRIX_3:5
.= ((M1 + M2) + (- M2)) + M3 by A1, A3, MATRIX_3:3
.= (M1 + M2) + ((- M2) + M3) by A7, A9, MATRIX_3:3
.= (M1 + M2) + ((- M2) + (- (- M3))) by Th1
.= (M1 + M2) - (M2 - M3) by A1, A3, A8, Th12 ;
:: thesis: verum

end;

suppose A10: len M1 = 0 ; :: thesis: M1 + M3 = (M1 + M2) - (M2 - M3)

A11: len ((M1 + M2) - (M2 - M3)) = len (M1 + M2) by MATRIX_3:def 3
.= len M1 by MATRIX_3:def 3 ;
len (M1 + M3) = len M1 by MATRIX_3:def 3;
hence M1 + M3 = (M1 + M2) - (M2 - M3) by A10, A11, CARD_2:64; :: thesis: verum

end;

end;