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

M2 - M1 is_less_than M2

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

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

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

A3: width M1 = width M2 by Lm3;

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

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

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

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

M2 - M1 is_less_than M2

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

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

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

A3: width M1 = width M2 by Lm3;

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

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

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

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

proof

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

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

then M1 * (i,j) > 0 by A1, A2, A4;

then (M2 * (i,j)) - (M1 * (i,j)) < M2 * (i,j) by XREAL_1:44;

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

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

then M1 * (i,j) > 0 by A1, A2, A4;

then (M2 * (i,j)) - (M1 * (i,j)) < M2 * (i,j) by XREAL_1:44;

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