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 )
A1: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;
assume A2: ( M1 is Positive & M2 is_less_or_equal_with M3 ) ; :: thesis: M2 is_less_than M1 + M3
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 ( M1 * (i,j) > 0 & M2 * (i,j) <= M3 * (i,j) ) by A2, A1;
then M2 * (i,j) < (M1 * (i,j)) + (M3 * (i,j)) by XREAL_1:39;
hence M2 * (i,j) < (M1 + M3) * (i,j) by A1, A3, MATRIXR1:25; :: thesis: verum

end;

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