let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n, REAL st M1 is Positive & M2 is_less_or_equal_with M3 holds
M2 is_less_than M1 + M3
let M1, M2, M3 be Matrix of n, REAL ; :: thesis: ( M1 is Positive & M2 is_less_or_equal_with M3 implies M2 is_less_than M1 + M3 )
assume A1: ( M1 is Positive & M2 is_less_or_equal_with M3 ) ; :: thesis: M2 is_less_than M1 + M3
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices M3 = [:(Seg n),(Seg n):] & Indices (M1 + M3) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
for i, j being Nat st [i,j] in Indices M2 holds
M2 * i,j < (M1 + M3) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M2 implies M2 * i,j < (M1 + M3) * i,j )
assume A3: [i,j] in Indices M2 ; :: thesis: M2 * i,j < (M1 + M3) * i,j
then A4: M1 * i,j > 0 by A1, A2, Def1;
M2 * i,j <= M3 * i,j by A1, A3, Def6;
then M2 * i,j < (M1 * i,j) + (M3 * i,j) by A4, XREAL_1:41;
hence M2 * i,j < (M1 + M3) * i,j by A2, A3, MATRIXR1:25; :: thesis: verum

end;

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