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

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M1) implies (a * M1) * (i,j) < (a * M2) * (i,j) )
assume A5: [i,j] in Indices (a * M1) ; :: thesis: (a * M1) * (i,j) < (a * M2) * (i,j)
then M1 * (i,j) < M2 * (i,j) by A1, A3;
then a * (M1 * (i,j)) < a * (M2 * (i,j)) by A2, XREAL_1:68;
then A6: (a * M1) * (i,j) < a * (M2 * (i,j)) by A3, A5, Th4;
[i,j] in Indices M2 by A4, A5, MATRIX_0:24;
hence (a * M1) * (i,j) < (a * M2) * (i,j) by A6, Th4; :: thesis: verum

end;

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