let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 & M1 - M2 = 0. (K,(len M1),(width M1)) holds

M1 = M2

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M1 - M2 = 0. (K,(len M1),(width M1)) implies M1 = M2 )

assume that

A1: len M1 = len M2 and

A2: width M1 = width M2 and

A3: M1 - M2 = 0. (K,(len M1),(width M1)) ; :: thesis: M1 = M2

M1 = M2

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M1 - M2 = 0. (K,(len M1),(width M1)) implies M1 = M2 )

assume that

A1: len M1 = len M2 and

A2: width M1 = width M2 and

A3: M1 - M2 = 0. (K,(len M1),(width M1)) ; :: thesis: M1 = M2

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

end;

suppose A4:
len M1 > 0
; :: thesis: M1 = M2

then A5:
M2 is Matrix of len M1, width M1,K
by A1, A2, MATRIX_0:20;

A6: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;

A7: len (0. (K,(len M1),(width M1))) = len M1 by MATRIX_0:def 2;

then width (0. (K,(len M1),(width M1))) = width M1 by A4, MATRIX_0:20;

then (M1 + (- M2)) + M2 = M2 + (0. (K,(len M1),(width M1))) by A1, A2, A3, A7, MATRIX_3:2

.= M2 by A5, MATRIX_3:4 ;

then M1 + ((- M2) + M2) = M2 by A1, A2, A6, MATRIX_3:3;

then M1 + (M2 + (- M2)) = M2 by A6, MATRIX_3:2;

then A8: M1 + (0. (K,(len M1),(width M1))) = M2 by A5, MATRIX_3:5;

M1 is Matrix of len M1, width M1,K by A4, MATRIX_0:20;

hence M1 = M2 by A8, MATRIX_3:4; :: thesis: verum

end;A6: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;

A7: len (0. (K,(len M1),(width M1))) = len M1 by MATRIX_0:def 2;

then width (0. (K,(len M1),(width M1))) = width M1 by A4, MATRIX_0:20;

then (M1 + (- M2)) + M2 = M2 + (0. (K,(len M1),(width M1))) by A1, A2, A3, A7, MATRIX_3:2

.= M2 by A5, MATRIX_3:4 ;

then M1 + ((- M2) + M2) = M2 by A1, A2, A6, MATRIX_3:3;

then M1 + (M2 + (- M2)) = M2 by A6, MATRIX_3:2;

then A8: M1 + (0. (K,(len M1),(width M1))) = M2 by A5, MATRIX_3:5;

M1 is Matrix of len M1, width M1,K by A4, MATRIX_0:20;

hence M1 = M2 by A8, MATRIX_3:4; :: thesis: verum