let K be Field; :: 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) - (M3 - M2)
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) - (M3 - M2) )
assume A1:
( len M1 = len M2 & len M2 = len M3 & width M1 = width M2 & width M2 = width M3 )
; :: thesis: M1 - M3 = (M1 - M2) - (M3 - M2)
A2:
( len (- M2) = len M2 & width (- M2) = width M2 )
by MATRIX_3:def 2;
A3:
( len (- M3) = len M3 & width (- M3) = width M3 )
by MATRIX_3:def 2;
per cases
( len M1 > 0 or len M1 = 0 )
by NAT_1:3;
suppose A4:
len M1 > 0
;
:: thesis: M1 - M3 = (M1 - M2) - (M3 - M2)then A5:
M2 is
Matrix of
len M1,
width M1,
K
by A1, MATRIX_1:20;
A6:
M1 is
Matrix of
len M1,
width M1,
K
by A4, MATRIX_1:20;
A7:
(
len (M1 + (- M2)) = len M1 &
width (M1 + (- M2)) = width M1 )
by MATRIX_3:def 3;
(M1 - M2) - (M3 - M2) =
(M1 + (- M2)) + ((- M3) + (- (- M2)))
by A1, A2, Th12
.=
(M1 + (- M2)) + ((- M3) + M2)
by Th1
.=
(M1 + (- M2)) + (M2 + (- M3))
by A1, A3, MATRIX_3:4
.=
((M1 + (- M2)) + M2) + (- M3)
by A1, A3, A7, MATRIX_3:5
.=
(M1 + ((- M2) + M2)) + (- M3)
by A1, A2, MATRIX_3:5
.=
(M1 + (M2 + (- M2))) + (- M3)
by A2, MATRIX_3:4
.=
(M1 + (0. K,(len M1),(width M1))) + (- M3)
by A5, MATRIX_3:7
.=
M1 + (- M3)
by A6, MATRIX_3:6
;
hence
M1 - M3 = (M1 - M2) - (M3 - M2)
;
:: thesis: verum end;