let a be 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 that
A1: M1 is_less_than M2 and
A2: a < 0 ; :: thesis: a * M2 is_less_than a * M1
A3: Indices (a * M2) = Indices M2 by MATRIXR1:28;
A4: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
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 A4, MATRIX_0:24;
then M1 * (i,j) < M2 * (i,j) by A1;
then a * (M2 * (i,j)) < a * (M1 * (i,j)) by A2, XREAL_1:69;
then (a * M2) * (i,j) < a * (M1 * (i,j)) by A3, A5, Th4;
hence (a * M2) * (i,j) < (a * M1) * (i,j) by A6, Th4; :: thesis: verum

end;

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