let a, b be Element of REAL ; :: thesis: for n being Nat
for M1, M2 being Matrix of n, REAL st a > 0 & a <= b & M1 is Positive & M1 is_less_than M2 holds
a * M1 is_less_than b * M2
let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st a > 0 & a <= b & M1 is Positive & M1 is_less_than M2 holds
a * M1 is_less_than b * M2
let M1, M2 be Matrix of n, REAL ; :: thesis: ( a > 0 & a <= b & M1 is Positive & M1 is_less_than M2 implies a * M1 is_less_than b * M2 )
assume that
A1: ( a > 0 & a <= b ) and
A2: ( M1 is Positive & M1 is_less_than M2 ) ; :: thesis: a * M1 is_less_than b * M2
A3: Indices (a * M1) = Indices M1 by MATRIXR1:28;
A4: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
for i, j being Nat st [i,j] in Indices (a * M1) holds
(a * M1) * i,j < (b * M2) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M1) implies (a * M1) * i,j < (b * M2) * i,j )
assume A5: [i,j] in Indices (a * M1) ; :: thesis: (a * M1) * i,j < (b * M2) * i,j
then ( M1 * i,j > 0 & M1 * i,j < M2 * i,j ) by A2, A3, Def1, Def5;
then a * (M1 * i,j) < b * (M2 * i,j) by A1, XREAL_1:100;
then (a * M1) * i,j < b * (M2 * i,j) by A3, A5, Th4;
hence (a * M1) * i,j < (b * M2) * i,j by A4, A3, A5, Th4; :: thesis: verum

end;

hence a * M1 is_less_than b * M2 by Def5; :: thesis: verum