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

M1 + M2 is Nonpositive

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

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

assume A4: M1 is_less_or_equal_with - M2 ; :: thesis: M1 + M2 is Nonpositive

for i, j being Nat st [i,j] in Indices (M1 + M2) holds

(M1 + M2) * (i,j) <= 0

M1 + M2 is Nonpositive

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

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

assume A4: M1 is_less_or_equal_with - M2 ; :: thesis: M1 + M2 is Nonpositive

for i, j being Nat st [i,j] in Indices (M1 + M2) holds

(M1 + M2) * (i,j) <= 0

proof

hence
M1 + M2 is Nonpositive
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) <= 0 )

assume A5: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) <= 0

then M1 * (i,j) <= (- M2) * (i,j) by A4, A1, A3;

then M1 * (i,j) <= - (M2 * (i,j)) by A2, A3, A5, Th2;

then (M1 * (i,j)) + (M2 * (i,j)) <= 0 by XREAL_1:59;

hence (M1 + M2) * (i,j) <= 0 by A1, A3, A5, MATRIXR1:25; :: thesis: verum

end;assume A5: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) <= 0

then M1 * (i,j) <= (- M2) * (i,j) by A4, A1, A3;

then M1 * (i,j) <= - (M2 * (i,j)) by A2, A3, A5, Th2;

then (M1 * (i,j)) + (M2 * (i,j)) <= 0 by XREAL_1:59;

hence (M1 + M2) * (i,j) <= 0 by A1, A3, A5, MATRIXR1:25; :: thesis: verum