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

M2 is_less_or_equal_with M2 - M1

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

assume A1: M1 is Nonpositive ; :: thesis: M2 is_less_or_equal_with M2 - M1

A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

A3: ( len M1 = len M2 & width M1 = width M2 ) by Lm3;

for i, j being Nat st [i,j] in Indices M2 holds

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

M2 is_less_or_equal_with M2 - M1

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

assume A1: M1 is Nonpositive ; :: thesis: M2 is_less_or_equal_with M2 - M1

A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

A3: ( len M1 = len M2 & width M1 = width M2 ) by Lm3;

for i, j being Nat st [i,j] in Indices M2 holds

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

proof

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

assume A4: [i,j] in Indices M2 ; :: thesis: M2 * (i,j) <= (M2 - M1) * (i,j)

then M1 * (i,j) <= 0 by A1, A2;

then M2 * (i,j) <= (M2 * (i,j)) - (M1 * (i,j)) by XREAL_1:45;

hence M2 * (i,j) <= (M2 - M1) * (i,j) by A3, A4, Th3; :: thesis: verum

end;assume A4: [i,j] in Indices M2 ; :: thesis: M2 * (i,j) <= (M2 - M1) * (i,j)

then M1 * (i,j) <= 0 by A1, A2;

then M2 * (i,j) <= (M2 * (i,j)) - (M1 * (i,j)) by XREAL_1:45;

hence M2 * (i,j) <= (M2 - M1) * (i,j) by A3, A4, Th3; :: thesis: verum