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

M1 - M2 is Positive

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

assume A1: M2 is_less_than M1 ; :: thesis: M1 - M2 is Positive

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

A3: width M1 = width M2 by Lm3;

A4: Indices (M1 - M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

A5: ( Indices M1 = [:(Seg n),(Seg n):] & len M1 = len M2 ) by Lm3, MATRIX_0:24;

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

(M1 - M2) * (i,j) > 0

M1 - M2 is Positive

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

assume A1: M2 is_less_than M1 ; :: thesis: M1 - M2 is Positive

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

A3: width M1 = width M2 by Lm3;

A4: Indices (M1 - M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

A5: ( Indices M1 = [:(Seg n),(Seg n):] & len M1 = len M2 ) by Lm3, MATRIX_0:24;

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

(M1 - M2) * (i,j) > 0

proof

hence
M1 - M2 is Positive
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 - M2) implies (M1 - M2) * (i,j) > 0 )

assume A6: [i,j] in Indices (M1 - M2) ; :: thesis: (M1 - M2) * (i,j) > 0

then M1 * (i,j) > M2 * (i,j) by A1, A2, A4;

then (M1 * (i,j)) - (M2 * (i,j)) > 0 by XREAL_1:50;

hence (M1 - M2) * (i,j) > 0 by A4, A5, A3, A6, Th3; :: thesis: verum

end;assume A6: [i,j] in Indices (M1 - M2) ; :: thesis: (M1 - M2) * (i,j) > 0

then M1 * (i,j) > M2 * (i,j) by A1, A2, A4;

then (M1 * (i,j)) - (M2 * (i,j)) > 0 by XREAL_1:50;

hence (M1 - M2) * (i,j) > 0 by A4, A5, A3, A6, Th3; :: thesis: verum