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_1:25;
A3: width M1 = width M2 by Lm1;
A4: Indices (M2 - M1) = [:(Seg n),(Seg n):] by MATRIX_1:25;
A5: ( Indices M2 = [:(Seg n),(Seg n):] & len M1 = len M2 ) by Lm1, MATRIX_1:25;
for i, j being Nat st [i,j] in Indices (M2 - M1) holds
(M2 - M1) * i,j < M2 * i,j

proof

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, Def1;
then (M2 * i,j) - (M1 * i,j) < M2 * i,j by XREAL_1:46;
hence (M2 - M1) * i,j < M2 * i,j by A4, A5, A3, A6, Th3; :: thesis: verum

end;

hence M2 - M1 is_less_than M2 by Def5; :: thesis: verum