let a be Element of REAL ; :: thesis: for n being Nat
for M1, M2 being Matrix of n, REAL st M1 is_less_than M2 & a < 0 holds
a * M2 is_less_than a * M1
let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st M1 is_less_than M2 & a < 0 holds
a * M2 is_less_than a * M1
let M1, M2 be Matrix of n, REAL ; :: thesis: ( M1 is_less_than M2 & a < 0 implies a * M2 is_less_than a * M1 )
assume A1: ( M1 is_less_than M2 & a < 0 ) ; :: thesis: a * M2 is_less_than a * M1
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 (a * M2) = len M2 & width (a * M2) = width M2 ) by MATRIXR1:27;
A4: ( Indices (a * M1) = Indices M1 & Indices (a * M2) = Indices M2 ) by MATRIXR1:28;
for i, j being Nat st [i,j] in Indices (a * M2) holds
(a * M2) * i,j < (a * M1) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M2) implies (a * M2) * i,j < (a * M1) * i,j )
assume A5: [i,j] in Indices (a * M2) ; :: thesis: (a * M2) * i,j < (a * M1) * i,j
then A6: [i,j] in Indices M1 by A2, MATRIXR1:28;
then M1 * i,j < M2 * i,j by A1, Def5;
then a * (M2 * i,j) < a * (M1 * i,j) by A1, XREAL_1:71;
then (a * M2) * i,j < a * (M1 * i,j) by A3, A4, A5, Th4;
hence (a * M2) * i,j < (a * M1) * i,j by A3, A6, Th4; :: thesis: verum

end;

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