let a, b, x, y be real number ; :: thesis: ( a >= 0 & x <= y implies (AffineMap a,b) . x <= (AffineMap a,b) . y )
assume ( a >= 0 & x <= y ) ; :: thesis: (AffineMap a,b) . x <= (AffineMap a,b) . y
then A1: a * x <= a * y by XREAL_1:66;
( (AffineMap a,b) . x = (a * x) + b & (AffineMap a,b) . y = (a * y) + b ) by Def3;
hence (AffineMap a,b) . x <= (AffineMap a,b) . y by A1, XREAL_1:9; :: thesis: verum