let n be Nat; for M1, M2, M3 being Matrix of n,REAL st M1 is Positive & M2 is_less_or_equal_with M3 holds
M2 is_less_than M1 + M3
let M1, M2, M3 be Matrix of n,REAL; ( M1 is Positive & M2 is_less_or_equal_with M3 implies M2 is_less_than M1 + M3 )
A1:
( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] )
by MATRIX_0:24;
assume A2:
( M1 is Positive & M2 is_less_or_equal_with M3 )
; M2 is_less_than M1 + M3
for i, j being Nat st [i,j] in Indices M2 holds
M2 * (i,j) < (M1 + M3) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices M2 implies M2 * (i,j) < (M1 + M3) * (i,j) )
assume A3:
[i,j] in Indices M2
;
M2 * (i,j) < (M1 + M3) * (i,j)
then
(
M1 * (
i,
j)
> 0 &
M2 * (
i,
j)
<= M3 * (
i,
j) )
by A2, A1;
then
M2 * (
i,
j)
< (M1 * (i,j)) + (M3 * (i,j))
by XREAL_1:39;
hence
M2 * (
i,
j)
< (M1 + M3) * (
i,
j)
by A1, A3, MATRIXR1:25;
verum
end;
hence
M2 is_less_than M1 + M3
; verum