let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| holds
M1 + M2 is Negative
let M1, M2 be Matrix of n, REAL ; :: thesis: ( M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| implies M1 + M2 is Negative )
assume A1: ( M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| ) ; :: thesis: M1 + M2 is Negative
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 ( M1 * i,j > 0 & M2 * i,j < 0 ) by A1, A2, Def1, Def2;
then ( abs (M1 * i,j) = M1 * i,j & - (M2 * i,j) = abs (M2 * i,j) ) by ABSVALUE:def 1;
then A4: (M1 * i,j) + (M2 * i,j) = (abs (M1 * i,j)) - (abs (M2 * i,j)) ;
[i,j] in Indices |:M1:| by A2, A3, Th5;
then |:M1:| * i,j < |:M2:| * i,j by A1, Def5;
then abs (M1 * i,j) < |:M2:| * i,j by A2, A3, Def7;
then abs (M1 * i,j) < abs (M2 * i,j) by A2, A3, Def7;
then (abs (M1 * i,j)) - (abs (M2 * i,j)) < (abs (M2 * i,j)) - (abs (M2 * i,j)) by XREAL_1:11;
hence (M1 + M2) * i,j < 0 by A2, A3, A4, MATRIXR1:25; :: thesis: verum

end;

hence M1 + M2 is Negative by Def2; :: thesis: verum