let n be Nat; for M1, M2, M3, M4 being Matrix of n,REAL st M1 + M2 is_less_or_equal_with M3 - M4 holds
M1 + M4 is_less_or_equal_with M3 - M2
let M1, M2, M3, M4 be Matrix of n,REAL; ( M1 + M2 is_less_or_equal_with M3 - M4 implies M1 + M4 is_less_or_equal_with M3 - M2 )
assume A1:
M1 + M2 is_less_or_equal_with M3 - M4
; M1 + M4 is_less_or_equal_with M3 - M2
A2:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
Indices (M1 + M2) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A4:
Indices (M1 + M4) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A5:
( len M2 = len M3 & width M2 = width M3 )
by Lm3;
A6:
Indices M3 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A7:
( len M3 = len M4 & width M3 = width M4 )
by Lm3;
for i, j being Nat st [i,j] in Indices (M1 + M4) holds
(M1 + M4) * (i,j) <= (M3 - M2) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M4) implies (M1 + M4) * (i,j) <= (M3 - M2) * (i,j) )
assume A8:
[i,j] in Indices (M1 + M4)
;
(M1 + M4) * (i,j) <= (M3 - M2) * (i,j)
then
(M1 + M2) * (
i,
j)
<= (M3 - M4) * (
i,
j)
by A1, A3, A4;
then
(M1 * (i,j)) + (M2 * (i,j)) <= (M3 - M4) * (
i,
j)
by A2, A4, A8, MATRIXR1:25;
then
(M1 * (i,j)) + (M2 * (i,j)) <= (M3 * (i,j)) - (M4 * (i,j))
by A6, A4, A7, A8, Th3;
then
(M1 * (i,j)) + (M4 * (i,j)) <= (M3 * (i,j)) - (M2 * (i,j))
by XREAL_1:22;
then
(M1 + M4) * (
i,
j)
<= (M3 * (i,j)) - (M2 * (i,j))
by A2, A4, A8, MATRIXR1:25;
hence
(M1 + M4) * (
i,
j)
<= (M3 - M2) * (
i,
j)
by A6, A4, A5, A8, Th3;
verum
end;
hence
M1 + M4 is_less_or_equal_with M3 - M2
; verum