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

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 - M4) implies (M1 - M4) * (i,j) < (M2 - M3) * (i,j) )
assume A8: [i,j] in Indices (M1 - M4) ; :: thesis: (M1 - M4) * (i,j) < (M2 - M3) * (i,j)
then ( M1 * (i,j) < M2 * (i,j) & M3 * (i,j) <= M4 * (i,j) ) by A1, A2, A4, A6;
then (M1 * (i,j)) - (M4 * (i,j)) < (M2 * (i,j)) - (M3 * (i,j)) by XREAL_1:14;
then (M1 - M4) * (i,j) < (M2 * (i,j)) - (M3 * (i,j)) by A2, A6, A7, A8, Th3;
hence (M1 - M4) * (i,j) < (M2 - M3) * (i,j) by A6, A3, A5, A8, Th3; :: thesis: verum

end;

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