let a, b be Real; :: thesis: for n being Nat
for M1, M2 being Matrix of n,REAL st a >= 0 & a <= b & M1 is Nonnegative & M1 is_less_or_equal_with M2 holds
a * M1 is_less_or_equal_with b * M2
let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st a >= 0 & a <= b & M1 is Nonnegative & M1 is_less_or_equal_with M2 holds
a * M1 is_less_or_equal_with b * M2
let M1, M2 be Matrix of n,REAL; :: thesis: ( a >= 0 & a <= b & M1 is Nonnegative & M1 is_less_or_equal_with M2 implies a * M1 is_less_or_equal_with b * M2 )
assume that
A1: ( a >= 0 & a <= b ) and
A2: ( M1 is Nonnegative & M1 is_less_or_equal_with M2 ) ; :: thesis: a * M1 is_less_or_equal_with 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_0:24;
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;
then a * (M1 * (i,j)) <= b * (M2 * (i,j)) by A1, XREAL_1:66;
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_or_equal_with b * M2 ; :: thesis: verum