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 )
A1: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A2: Indices M3 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: ( Indices M1 = [:(Seg n),(Seg n):] & Indices (M1 + M3) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;
assume A4: ( M1 is_less_than M2 & M3 is_less_than M4 ) ; :: thesis: M1 + M3 is_less_than M2 + M4
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 A5: [i,j] in Indices (M1 + M3) ; :: thesis: (M1 + M3) * (i,j) < (M2 + M4) * (i,j)
then A6: ( (M1 + M3) * (i,j) = (M1 * (i,j)) + (M3 * (i,j)) & (M2 * (i,j)) + (M4 * (i,j)) = (M2 + M4) * (i,j) ) by A1, A3, MATRIXR1:25;
( M1 * (i,j) < M2 * (i,j) & M3 * (i,j) < M4 * (i,j) ) by A4, A2, A3, A5;
hence (M1 + M3) * (i,j) < (M2 + M4) * (i,j) by A6, XREAL_1:8; :: thesis: verum

end;

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