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
  [!(r1(#)f1+r2(#)f2),x0,x1!] = r1*[!f1,x0,x1!] + r2*[!f2,x0,x1!]
proof
  [!(r1(#)f1+r2(#)f2),x0,x1!] = [!(r1(#)f1),x0,x1!]+[!(r2(#)f2),x0,x1!] by Th32
    .= r1*[!f1,x0,x1!]+[!(r2(#)f2),x0,x1!] by Th31
    .= r1*[!f1,x0,x1!]+r2*[!f2,x0,x1!] by Th31;
  hence thesis;
end;
