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

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M2 implies M2 * i,j < (M2 + M1) * i,j )
assume A3: [i,j] in Indices M2 ; :: thesis: M2 * i,j < (M2 + M1) * i,j
then M1 * i,j > 0 by A1, A2, Def1;
then M2 * i,j < (M2 * i,j) + (M1 * i,j) by XREAL_1:31;
hence M2 * i,j < (M2 + M1) * i,j by A3, MATRIXR1:25; :: thesis: verum

end;

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