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

M2 is_less_than M3 - M1

let M1, M2, M3 be Matrix of n,REAL; :: thesis: ( M1 is Nonpositive & M2 is_less_than M3 implies M2 is_less_than M3 - M1 )

assume A1: ( M1 is Nonpositive & M2 is_less_than M3 ) ; :: thesis: M2 is_less_than M3 - M1

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

A3: width M2 = width M3 by Lm3;

A4: ( width M1 = width M2 & len M2 = len M3 ) by Lm3;

A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A6: ( Indices M3 = [:(Seg n),(Seg n):] & len M1 = len M2 ) by Lm3, MATRIX_0:24;

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

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

M2 is_less_than M3 - M1

let M1, M2, M3 be Matrix of n,REAL; :: thesis: ( M1 is Nonpositive & M2 is_less_than M3 implies M2 is_less_than M3 - M1 )

assume A1: ( M1 is Nonpositive & M2 is_less_than M3 ) ; :: thesis: M2 is_less_than M3 - M1

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

A3: width M2 = width M3 by Lm3;

A4: ( width M1 = width M2 & len M2 = len M3 ) by Lm3;

A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A6: ( Indices M3 = [:(Seg n),(Seg n):] & len M1 = len M2 ) by Lm3, MATRIX_0:24;

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

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

proof

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

assume A7: [i,j] in Indices M2 ; :: thesis: M2 * (i,j) < (M3 - M1) * (i,j)

then ( M1 * (i,j) <= 0 & M2 * (i,j) < M3 * (i,j) ) by A1, A2, A5;

then M2 * (i,j) < (M3 * (i,j)) - (M1 * (i,j)) by XREAL_1:53;

hence M2 * (i,j) < (M3 - M1) * (i,j) by A5, A6, A4, A3, A7, Th3; :: thesis: verum

end;assume A7: [i,j] in Indices M2 ; :: thesis: M2 * (i,j) < (M3 - M1) * (i,j)

then ( M1 * (i,j) <= 0 & M2 * (i,j) < M3 * (i,j) ) by A1, A2, A5;

then M2 * (i,j) < (M3 * (i,j)) - (M1 * (i,j)) by XREAL_1:53;

hence M2 * (i,j) < (M3 - M1) * (i,j) by A5, A6, A4, A3, A7, Th3; :: thesis: verum