let n be Nat; :: thesis: for M1, M2, M3, M4 being Matrix of n, REAL st M1 is_less_than M2 & M3 is_less_than M4 holds
M1 + M3 is_less_than M2 + M4
let M1, M2, M3, M4 be Matrix of n, REAL ; :: thesis: ( M1 is_less_than M2 & M3 is_less_than M4 implies M1 + M3 is_less_than M2 + M4 )
assume A1: ( M1 is_less_than M2 & M3 is_less_than M4 ) ; :: thesis: M1 + M3 is_less_than M2 + M4
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices M3 = [:(Seg n),(Seg n):] & Indices M4 = [:(Seg n),(Seg n):] & Indices (M1 + M3) = [:(Seg n),(Seg n):] & Indices (M2 + M4) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
for i, j being Nat st [i,j] in Indices (M1 + M3) holds
(M1 + M3) * i,j < (M2 + M4) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M3) implies (M1 + M3) * i,j < (M2 + M4) * i,j )
assume A3: [i,j] in Indices (M1 + M3) ; :: thesis: (M1 + M3) * i,j < (M2 + M4) * i,j
then A4: ( (M1 + M3) * i,j = (M1 * i,j) + (M3 * i,j) & (M2 * i,j) + (M4 * i,j) = (M2 + M4) * i,j ) by A2, MATRIXR1:25;
A5: M1 * i,j < M2 * i,j by A1, A2, A3, Def5;
M3 * i,j < M4 * i,j by A1, A2, A3, Def5;
hence (M1 + M3) * i,j < (M2 + M4) * i,j by A4, A5, XREAL_1:10; :: thesis: verum

end;

hence M1 + M3 is_less_than M2 + M4 by Def5; :: thesis: verum