reserve n,m,k,i for Nat,
  h,r,r1,r2,x0,x1,x2,x for Real,
  S for Functional_Sequence of REAL,REAL,
  y for set;
reserve f,f1,f2 for Function of REAL,REAL;

theorem Th31:
  [!(r(#)f),x0,x1!] = r*[!f,x0,x1!]
proof
  x1 in REAL by XREAL_0:def 1;
  then
A1: x1 in dom (r(#)f) by FUNCT_2:def 1;
  x0 in REAL by XREAL_0:def 1;
  then x0 in dom (r(#)f) by FUNCT_2:def 1;
  then [!(r(#)f),x0,x1!] = (r*f.x0-(r(#)f).x1)/(x0-x1) by VALUED_1:def 5
    .= (r*f.x0-r*f.x1)/(x0-x1) by A1,VALUED_1:def 5
    .= r*(f.x0-f.x1)/(x0-x1)
    .= r*[!f,x0,x1!] by XCMPLX_1:74;
  hence thesis;
end;
