let n be Nat; :: thesis: for M1, M2, M3, M4 being Matrix of n,REAL st M1 + M2 is_less_or_equal_with M3 - M4 holds
M1 + M4 is_less_or_equal_with M3 - M2

let M1, M2, M3, M4 be Matrix of n,REAL; :: thesis: ( M1 + M2 is_less_or_equal_with M3 - M4 implies M1 + M4 is_less_or_equal_with M3 - M2 )
assume A1: M1 + M2 is_less_or_equal_with M3 - M4 ; :: thesis: M1 + M4 is_less_or_equal_with M3 - M2
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A4: Indices (M1 + M4) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A5: ( len M2 = len M3 & width M2 = width M3 ) by Lm3;
A6: Indices M3 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A7: ( len M3 = len M4 & width M3 = width M4 ) by Lm3;
for i, j being Nat st [i,j] in Indices (M1 + M4) holds
(M1 + M4) * (i,j) <= (M3 - M2) * (i,j)
proof
let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M4) implies (M1 + M4) * (i,j) <= (M3 - M2) * (i,j) )
assume A8: [i,j] in Indices (M1 + M4) ; :: thesis: (M1 + M4) * (i,j) <= (M3 - M2) * (i,j)
then (M1 + M2) * (i,j) <= (M3 - M4) * (i,j) by A1, A3, A4;
then (M1 * (i,j)) + (M2 * (i,j)) <= (M3 - M4) * (i,j) by ;
then (M1 * (i,j)) + (M2 * (i,j)) <= (M3 * (i,j)) - (M4 * (i,j)) by A6, A4, A7, A8, Th3;
then (M1 * (i,j)) + (M4 * (i,j)) <= (M3 * (i,j)) - (M2 * (i,j)) by XREAL_1:22;
then (M1 + M4) * (i,j) <= (M3 * (i,j)) - (M2 * (i,j)) by ;
hence (M1 + M4) * (i,j) <= (M3 - M2) * (i,j) by A6, A4, A5, A8, Th3; :: thesis: verum
end;
hence M1 + M4 is_less_or_equal_with M3 - M2 ; :: thesis: verum