let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n,REAL st M1 is Negative & M3 is_less_or_equal_with M2 holds

M3 + M1 is_less_than M2

let M1, M2, M3 be Matrix of n,REAL; :: thesis: ( M1 is Negative & M3 is_less_or_equal_with M2 implies M3 + M1 is_less_than M2 )

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

A2: ( Indices M3 = [:(Seg n),(Seg n):] & Indices (M3 + M1) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A3: ( M1 is Negative & M3 is_less_or_equal_with M2 ) ; :: thesis: M3 + M1 is_less_than M2

for i, j being Nat st [i,j] in Indices (M3 + M1) holds

(M3 + M1) * (i,j) < M2 * (i,j)

M3 + M1 is_less_than M2

let M1, M2, M3 be Matrix of n,REAL; :: thesis: ( M1 is Negative & M3 is_less_or_equal_with M2 implies M3 + M1 is_less_than M2 )

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

A2: ( Indices M3 = [:(Seg n),(Seg n):] & Indices (M3 + M1) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A3: ( M1 is Negative & M3 is_less_or_equal_with M2 ) ; :: thesis: M3 + M1 is_less_than M2

for i, j being Nat st [i,j] in Indices (M3 + M1) holds

(M3 + M1) * (i,j) < M2 * (i,j)

proof

hence
M3 + M1 is_less_than M2
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M3 + M1) implies (M3 + M1) * (i,j) < M2 * (i,j) )

assume A4: [i,j] in Indices (M3 + M1) ; :: thesis: (M3 + M1) * (i,j) < M2 * (i,j)

then ( M1 * (i,j) < 0 & M3 * (i,j) <= M2 * (i,j) ) by A3, A1, A2;

then (M3 * (i,j)) + (M1 * (i,j)) < M2 * (i,j) by XREAL_1:37;

hence (M3 + M1) * (i,j) < M2 * (i,j) by A2, A4, MATRIXR1:25; :: thesis: verum

end;assume A4: [i,j] in Indices (M3 + M1) ; :: thesis: (M3 + M1) * (i,j) < M2 * (i,j)

then ( M1 * (i,j) < 0 & M3 * (i,j) <= M2 * (i,j) ) by A3, A1, A2;

then (M3 * (i,j)) + (M1 * (i,j)) < M2 * (i,j) by XREAL_1:37;

hence (M3 + M1) * (i,j) < M2 * (i,j) by A2, A4, MATRIXR1:25; :: thesis: verum