let n be Nat; for M1, M2, M3 being Matrix of n,REAL st M1 is_less_or_equal_with M2 holds
M1 - M3 is_less_or_equal_with M2 - M3
let M1, M2, M3 be Matrix of n,REAL; ( M1 is_less_or_equal_with M2 implies M1 - M3 is_less_or_equal_with M2 - M3 )
assume A1:
M1 is_less_or_equal_with M2
; M1 - M3 is_less_or_equal_with M2 - M3
A2:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
width M2 = width M3
by Lm3;
A4:
( Indices M2 = [:(Seg n),(Seg n):] & len M2 = len M3 )
by Lm3, MATRIX_0:24;
A5:
Indices (M1 - M3) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A6:
( len M1 = len M3 & width M1 = width M3 )
by Lm3;
for i, j being Nat st [i,j] in Indices (M1 - M3) holds
(M1 - M3) * (i,j) <= (M2 - M3) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 - M3) implies (M1 - M3) * (i,j) <= (M2 - M3) * (i,j) )
assume A7:
[i,j] in Indices (M1 - M3)
;
(M1 - M3) * (i,j) <= (M2 - M3) * (i,j)
then
M1 * (
i,
j)
<= M2 * (
i,
j)
by A1, A2, A5;
then
(M1 * (i,j)) - (M3 * (i,j)) <= (M2 * (i,j)) - (M3 * (i,j))
by XREAL_1:9;
then
(M1 - M3) * (
i,
j)
<= (M2 * (i,j)) - (M3 * (i,j))
by A2, A5, A6, A7, Th3;
hence
(M1 - M3) * (
i,
j)
<= (M2 - M3) * (
i,
j)
by A5, A4, A3, A7, Th3;
verum
end;
hence
M1 - M3 is_less_or_equal_with M2 - M3
; verum