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

- M1 is_less_or_equal_with M2

let M1, M2 be Matrix of n,REAL; :: thesis: ( - M2 is_less_or_equal_with M1 implies - M1 is_less_or_equal_with M2 )

A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A2: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A3: Indices (- M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

A4: Indices (- M1) = [:(Seg n),(Seg n):] by MATRIX_0:24;

assume A5: - M2 is_less_or_equal_with M1 ; :: thesis: - M1 is_less_or_equal_with M2

for i, j being Nat st [i,j] in Indices (- M1) holds

(- M1) * (i,j) <= M2 * (i,j)

- M1 is_less_or_equal_with M2

let M1, M2 be Matrix of n,REAL; :: thesis: ( - M2 is_less_or_equal_with M1 implies - M1 is_less_or_equal_with M2 )

A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A2: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A3: Indices (- M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

A4: Indices (- M1) = [:(Seg n),(Seg n):] by MATRIX_0:24;

assume A5: - M2 is_less_or_equal_with M1 ; :: thesis: - M1 is_less_or_equal_with M2

for i, j being Nat st [i,j] in Indices (- M1) holds

(- M1) * (i,j) <= M2 * (i,j)

proof

hence
- M1 is_less_or_equal_with M2
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (- M1) implies (- M1) * (i,j) <= M2 * (i,j) )

assume A6: [i,j] in Indices (- M1) ; :: thesis: (- M1) * (i,j) <= M2 * (i,j)

then (- M2) * (i,j) <= M1 * (i,j) by A5, A4, A3;

then - (M2 * (i,j)) <= M1 * (i,j) by A2, A4, A6, Th2;

then - (M1 * (i,j)) <= M2 * (i,j) by XREAL_1:26;

hence (- M1) * (i,j) <= M2 * (i,j) by A1, A4, A6, Th2; :: thesis: verum

end;assume A6: [i,j] in Indices (- M1) ; :: thesis: (- M1) * (i,j) <= M2 * (i,j)

then (- M2) * (i,j) <= M1 * (i,j) by A5, A4, A3;

then - (M2 * (i,j)) <= M1 * (i,j) by A2, A4, A6, Th2;

then - (M1 * (i,j)) <= M2 * (i,j) by XREAL_1:26;

hence (- M1) * (i,j) <= M2 * (i,j) by A1, A4, A6, Th2; :: thesis: verum