let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st - M2 is_less_than M1 holds
M1 + M2 is Positive
let M1, M2 be Matrix of n,REAL; :: thesis: ( - M2 is_less_than M1 implies M1 + M2 is Positive )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A2: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: Indices (- M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A4: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
assume A5: - M2 is_less_than M1 ; :: thesis: M1 + M2 is Positive
for i, j being Nat st [i,j] in Indices (M1 + M2) holds
(M1 + M2) * (i,j) > 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) > 0 )
assume A6: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) > 0
then (- M2) * (i,j) < M1 * (i,j) by A5, A3, A4;
then - (M2 * (i,j)) < M1 * (i,j) by A2, A4, A6, Th2;
then (M1 * (i,j)) + (M2 * (i,j)) > 0 by XREAL_1:62;
hence (M1 + M2) * (i,j) > 0 by A1, A4, A6, MATRIXR1:25; :: thesis: verum

end;

hence M1 + M2 is Positive ; :: thesis: verum