let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n, REAL st M1 is_less_than M2 holds
M3 - M2 is_less_than M3 - M1
let M1, M2, M3 be Matrix of n, REAL ; :: thesis: ( M1 is_less_than M2 implies M3 - M2 is_less_than M3 - M1 )
assume A1: M1 is_less_than M2 ; :: thesis: M3 - M2 is_less_than M3 - M1
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_1:25;
A3: Indices (M3 - M1) = [:(Seg n),(Seg n):] by MATRIX_1:25;
A4: width M2 = width M3 by Lm1;
A5: ( Indices M3 = [:(Seg n),(Seg n):] & len M2 = len M3 ) by Lm1, MATRIX_1:25;
A6: ( len M1 = len M2 & width M1 = width M2 ) by Lm1;
A7: for i, j being Nat st [i,j] in Indices (M3 - M1) holds
(M3 - M2) * i,j < (M3 - M1) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M3 - M1) implies (M3 - M2) * i,j < (M3 - M1) * i,j )
assume A8: [i,j] in Indices (M3 - M1) ; :: thesis: (M3 - M2) * i,j < (M3 - M1) * i,j
then M1 * i,j < M2 * i,j by A1, A2, A3, Def5;
then (M3 * i,j) - (M2 * i,j) < (M3 * i,j) - (M1 * i,j) by XREAL_1:17;
then (M3 - M2) * i,j < (M3 * i,j) - (M1 * i,j) by A3, A5, A4, A8, Th3;
hence (M3 - M2) * i,j < (M3 - M1) * i,j by A3, A6, A5, A4, A8, Th3; :: thesis: verum

end;

Indices (M3 - M2) = [:(Seg n),(Seg n):] by MATRIX_1:25;
hence M3 - M2 is_less_than M3 - M1 by A3, A7, Def5; :: thesis: verum