let n be Nat; for M1, M2, M3 being Matrix of n,REAL st M1 is Nonpositive & M2 is_less_than M3 holds
M2 is_less_than M3 - M1
let M1, M2, M3 be Matrix of n,REAL; ( M1 is Nonpositive & M2 is_less_than M3 implies M2 is_less_than M3 - M1 )
assume A1:
( M1 is Nonpositive & M2 is_less_than M3 )
; M2 is_less_than M3 - M1
A2:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
width M2 = width M3
by Lm3;
A4:
( width M1 = width M2 & len M2 = len M3 )
by Lm3;
A5:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A6:
( Indices M3 = [:(Seg n),(Seg n):] & len M1 = len M2 )
by Lm3, MATRIX_0:24;
for i, j being Nat st [i,j] in Indices M2 holds
M2 * (i,j) < (M3 - M1) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices M2 implies M2 * (i,j) < (M3 - M1) * (i,j) )
assume A7:
[i,j] in Indices M2
;
M2 * (i,j) < (M3 - M1) * (i,j)
then
(
M1 * (
i,
j)
<= 0 &
M2 * (
i,
j)
< M3 * (
i,
j) )
by A1, A2, A5;
then
M2 * (
i,
j)
< (M3 * (i,j)) - (M1 * (i,j))
by XREAL_1:53;
hence
M2 * (
i,
j)
< (M3 - M1) * (
i,
j)
by A5, A6, A4, A3, A7, Th3;
verum
end;
hence
M2 is_less_than M3 - M1
; verum