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):] by MATRIX_1:25;
A3: ( Indices M2 = [:(Seg n),(Seg n):] & Indices (M2 - M1) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
A4: ( len M1 = len M2 & width M1 = width M2 ) 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 A5: [i,j] in Indices (M2 - M1) ; :: thesis: (M2 - M1) * i,j < M3 * i,j
then ( M1 * i,j >= 0 & M2 * i,j < M3 * i,j ) by A1, A2, A3, Def4, Def5;
then (M2 * i,j) - (M1 * i,j) < M3 * i,j by XREAL_1:53;
hence (M2 - M1) * i,j < M3 * i,j by A3, A4, A5, Th3; :: thesis: verum

end;

hence M2 - M1 is_less_than M3 by Def5; :: thesis: verum