let X be LinearTopSpace; :: thesis: for r being non zero Real holds mlt (r,X) is one-to-one
let r be non zero Real; :: thesis: mlt (r,X) is one-to-one
now :: thesis: for x1, x2 being object st x1 in the carrier of X & x2 in the carrier of X & (mlt (r,X)) . x1 = (mlt (r,X)) . x2 holds
x1 = x2
let x1, x2 be object ; :: thesis: ( x1 in the carrier of X & x2 in the carrier of X & (mlt (r,X)) . x1 = (mlt (r,X)) . x2 implies x1 = x2 )
assume that
A1: ( x1 in the carrier of X & x2 in the carrier of X ) and
A2: (mlt (r,X)) . x1 = (mlt (r,X)) . x2 ; :: thesis: x1 = x2
reconsider x19 = x1, x29 = x2 as Point of X by A1;
( (mlt (r,X)) . x1 = r * x19 & (mlt (r,X)) . x2 = r * x29 ) by Def13;
hence x1 = x2 by A2, RLVECT_1:36; :: thesis: verum
end;
hence mlt (r,X) is one-to-one by FUNCT_2:19; :: thesis: verum