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_or_equal_with 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_or_equal_with 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_or_equal_with M2 implies a * M1 is_less_than b * M2 )
assume A1: ( a >= 0 & a < b & M1 is Positive & M1 is_less_or_equal_with M2 ) ; :: thesis: a * M1 is_less_than b * M2
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
A3: ( len (a * M1) = len M1 & width (a * M1) = width M1 & len (b * M2) = len M2 & width (b * M2) = width M2 ) by MATRIXR1:27;
A4: ( Indices (a * M1) = Indices M1 & Indices (b * M2) = Indices M2 ) by MATRIXR1:28;
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 A6: M1 * i,j > 0 by A1, A4, Def1;
M1 * i,j <= M2 * i,j by A1, A4, A5, Def6;
then a * (M1 * i,j) < b * (M2 * i,j) by A1, A6, XREAL_1:99;
then (a * M1) * i,j < b * (M2 * i,j) by A3, A4, A5, Th4;
hence (a * M1) * i,j < (b * M2) * i,j by A2, A3, A4, A5, Th4; :: thesis: verum

end;

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