let n be Nat; :: thesis: 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 ; :: thesis: ( 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 ) ; :: thesis: M2 is_less_than M3 - M1
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_1:25;
A3: width M2 = width M3 by Lm1;
A4: ( width M1 = width M2 & len M2 = len M3 ) by Lm1;
A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_1:25;
A6: ( Indices M3 = [:(Seg n),(Seg n):] & len M1 = len M2 ) by Lm1, MATRIX_1:25;
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; :: thesis: ( [i,j] in Indices M2 implies M2 * i,j < (M3 - M1) * i,j )
assume A7: [i,j] in Indices M2 ; :: thesis: M2 * i,j < (M3 - M1) * i,j
then ( M1 * i,j <= 0 & M2 * i,j < M3 * i,j ) by A1, A2, A5, Def3, Def5;
then M2 * i,j < (M3 * i,j) - (M1 * i,j) by XREAL_1:55;
hence M2 * i,j < (M3 - M1) * i,j by A5, A6, A4, A3, A7, Th3; :: thesis: verum

end;

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