let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n,REAL st M1 + M2 is_less_or_equal_with M3 holds
M1 is_less_or_equal_with M3 - M2
let M1, M2, M3 be Matrix of n,REAL; :: thesis: ( M1 + M2 is_less_or_equal_with M3 implies M1 is_less_or_equal_with M3 - M2 )
assume A1: M1 + M2 is_less_or_equal_with M3 ; :: thesis: M1 is_less_or_equal_with M3 - M2
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: width M2 = width M3 by Lm3;
A4: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A5: ( Indices M3 = [:(Seg n),(Seg n):] & len M2 = len M3 ) by Lm3, MATRIX_0:24;
for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) <= (M3 - M2) * (i,j)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) <= (M3 - M2) * (i,j) )
assume A6: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) <= (M3 - M2) * (i,j)
then (M1 + M2) * (i,j) <= M3 * (i,j) by A1, A2, A4;
then (M1 * (i,j)) + (M2 * (i,j)) <= M3 * (i,j) by A6, MATRIXR1:25;
then M1 * (i,j) <= (M3 * (i,j)) - (M2 * (i,j)) by XREAL_1:19;
hence M1 * (i,j) <= (M3 - M2) * (i,j) by A2, A5, A3, A6, Th3; :: thesis: verum

end;

hence M1 is_less_or_equal_with M3 - M2 ; :: thesis: verum