let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st M1 is Positive & M2 is Negative & |:M2:| is_less_than |:M1:| holds
M1 + M2 is Positive
let M1, M2 be Matrix of n, REAL ; :: thesis: ( M1 is Positive & M2 is Negative & |:M2:| is_less_than |:M1:| implies M1 + M2 is Positive )
assume A1: ( M1 is Positive & M2 is Negative & |:M2:| is_less_than |:M1:| ) ; :: thesis: M1 + M2 is Positive
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 (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 A3: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * i,j > 0
then A4: M1 * i,j > 0 by A1, A2, Def1;
A5: M2 * i,j < 0 by A1, A2, A3, Def2;
A6: M1 * i,j = abs (M1 * i,j) by A4, ABSVALUE:def 1;
A7: - (M2 * i,j) = abs (M2 * i,j) by A5, ABSVALUE:def 1;
[i,j] in Indices |:M2:| by A2, A3, Th5;
then |:M2:| * i,j < |:M1:| * i,j by A1, Def5;
then abs (M2 * i,j) < |:M1:| * i,j by A2, A3, Def7;
then abs (M2 * i,j) < abs (M1 * i,j) by A2, A3, Def7;
then (abs (M1 * i,j)) - (abs (M2 * i,j)) > 0 by XREAL_1:52;
then (M1 * i,j) + (M2 * i,j) > 0 by A6, A7;
hence (M1 + M2) * i,j > 0 by A2, A3, MATRIXR1:25; :: thesis: verum

end;

hence M1 + M2 is Positive by Def1; :: thesis: verum