let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n,REAL st M1 is_less_than M2 holds
M1 + M3 is_less_than M2 + M3
let M1, M2, M3 be Matrix of n,REAL; :: thesis: ( M1 is_less_than M2 implies M1 + M3 is_less_than M2 + M3 )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A2: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: Indices (M1 + M3) = [:(Seg n),(Seg n):] by MATRIX_0:24;
assume A4: M1 is_less_than M2 ; :: thesis: M1 + M3 is_less_than M2 + M3
for i, j being Nat st [i,j] in Indices (M1 + M3) holds
(M1 + M3) * (i,j) < (M2 + M3) * (i,j)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M3) implies (M1 + M3) * (i,j) < (M2 + M3) * (i,j) )
assume A5: [i,j] in Indices (M1 + M3) ; :: thesis: (M1 + M3) * (i,j) < (M2 + M3) * (i,j)
then M1 * (i,j) < M2 * (i,j) by A4, A1, A3;
then (M1 * (i,j)) + (M3 * (i,j)) < (M2 * (i,j)) + (M3 * (i,j)) by XREAL_1:8;
then (M1 + M3) * (i,j) < (M2 * (i,j)) + (M3 * (i,j)) by A1, A3, A5, MATRIXR1:25;
hence (M1 + M3) * (i,j) < (M2 + M3) * (i,j) by A2, A3, A5, MATRIXR1:25; :: thesis: verum

end;

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