let n be Nat; for M1, M2 being Matrix of n,REAL st M1 is_less_or_equal_with - M2 holds
M1 + M2 is Nonpositive
let M1, M2 be Matrix of n,REAL; ( M1 is_less_or_equal_with - M2 implies M1 + M2 is Nonpositive )
A1:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A2:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
Indices (M1 + M2) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
assume A4:
M1 is_less_or_equal_with - M2
; M1 + M2 is Nonpositive
for i, j being Nat st [i,j] in Indices (M1 + M2) holds
(M1 + M2) * (i,j) <= 0
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) <= 0 )
assume A5:
[i,j] in Indices (M1 + M2)
;
(M1 + M2) * (i,j) <= 0
then
M1 * (
i,
j)
<= (- M2) * (
i,
j)
by A4, A1, A3;
then
M1 * (
i,
j)
<= - (M2 * (i,j))
by A2, A3, A5, Th2;
then
(M1 * (i,j)) + (M2 * (i,j)) <= 0
by XREAL_1:59;
hence
(M1 + M2) * (
i,
j)
<= 0
by A1, A3, A5, MATRIXR1:25;
verum
end;
hence
M1 + M2 is Nonpositive
; verum