let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n, REAL st M1 is Nonnegative & M2 is_less_than M3 holds
M2 - M1 is_less_than M3
let M1, M2, M3 be Matrix of n, REAL ; :: thesis: ( M1 is Nonnegative & M2 is_less_than M3 implies M2 - M1 is_less_than M3 )
assume A1:
( M1 is Nonnegative & M2 is_less_than M3 )
; :: thesis: M2 - M1 is_less_than M3
A2:
( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices M3 = [:(Seg n),(Seg n):] & Indices (M2 - M1) = [:(Seg n),(Seg n):] )
by MATRIX_1:25;
A3:
( len M1 = len M2 & width M1 = width M2 & len M2 = len M3 & width M2 = width M3 )
by Lm1;
for i, j being Nat st [i,j] in Indices (M2 - M1) holds
(M2 - M1) * i,j < M3 * i,j
proof
let i,
j be
Nat;
:: thesis: ( [i,j] in Indices (M2 - M1) implies (M2 - M1) * i,j < M3 * i,j )
assume A4:
[i,j] in Indices (M2 - M1)
;
:: thesis: (M2 - M1) * i,j < M3 * i,j
then A5:
M1 * i,
j >= 0
by A1, A2, Def4;
M2 * i,
j < M3 * i,
j
by A1, A2, A4, Def5;
then
(M2 * i,j) - (M1 * i,j) < M3 * i,
j
by A5, XREAL_1:53;
hence
(M2 - M1) * i,
j < M3 * i,
j
by A2, A3, A4, Th3;
:: thesis: verum
end;
hence
M2 - M1 is_less_than M3
by Def5; :: thesis: verum