let n be Nat; for M1, M2, M3 being Matrix of n,REAL st M1 is Negative & M3 is_less_or_equal_with M2 holds
M3 + M1 is_less_than M2
let M1, M2, M3 be Matrix of n,REAL; ( M1 is Negative & M3 is_less_or_equal_with M2 implies M3 + M1 is_less_than M2 )
A1:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A2:
( Indices M3 = [:(Seg n),(Seg n):] & Indices (M3 + M1) = [:(Seg n),(Seg n):] )
by MATRIX_0:24;
assume A3:
( M1 is Negative & M3 is_less_or_equal_with M2 )
; M3 + M1 is_less_than M2
for i, j being Nat st [i,j] in Indices (M3 + M1) holds
(M3 + M1) * (i,j) < M2 * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (M3 + M1) implies (M3 + M1) * (i,j) < M2 * (i,j) )
assume A4:
[i,j] in Indices (M3 + M1)
;
(M3 + M1) * (i,j) < M2 * (i,j)
then
(
M1 * (
i,
j)
< 0 &
M3 * (
i,
j)
<= M2 * (
i,
j) )
by A3, A1, A2;
then
(M3 * (i,j)) + (M1 * (i,j)) < M2 * (
i,
j)
by XREAL_1:37;
hence
(M3 + M1) * (
i,
j)
< M2 * (
i,
j)
by A2, A4, MATRIXR1:25;
verum
end;
hence
M3 + M1 is_less_than M2
; verum