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) - (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 A1:
( len M1 = len M2 & len M2 = len M3 & width M1 = width M2 & 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 A2:
len M1 > 0
;
:: thesis: M1 + M3 = (M1 + M2) - (M2 - M3)then A3:
M1 is
Matrix of
len M1,
width M1,
K
by MATRIX_1:20;
A4:
M2 is
Matrix of
len M1,
width M1,
K
by A1, A2, MATRIX_1:20;
A5:
(
len (- M2) = len M1 &
width (- M2) = width M1 )
by A1, MATRIX_3:def 2;
A6:
(
len (- M3) = len M1 &
width (- M3) = width M1 )
by A1, MATRIX_3:def 2;
A7:
(
len (M1 + M2) = len M1 &
width (M1 + M2) = width M1 )
by MATRIX_3:def 3;
thus M1 + M3 =
(M1 + (0. K,(len M1),(width M1))) + M3
by A3, MATRIX_3:6
.=
(M1 + (M2 + (- M2))) + M3
by A4, MATRIX_3:7
.=
((M1 + M2) + (- M2)) + M3
by A1, A5, MATRIX_3:5
.=
(M1 + M2) + ((- M2) + M3)
by A1, A5, A7, MATRIX_3:5
.=
(M1 + M2) + ((- M2) + (- (- M3)))
by Th1
.=
(M1 + M2) - (M2 - M3)
by A1, A6, Th12
;
:: thesis: verum end;