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
  fdif(r1(#)f1+r2(#)f2,h).(n+1).x = r1* fdif(f1,h).(n+1).x + r2* fdif(f2
  ,h).(n+1).x
proof
  set g1=r1(#)f1;
  set g2=r2(#)f2;
  fdif(r1(#)f1+r2(#)f2,h).(n+1).x = fdif(g1,h).(n+1).x + fdif(g2,h).(n+1).
  x by Th8
    .= r1* fdif(f1,h).(n+1).x + fdif(g2,h).(n+1).x by Th7
    .= r1* fdif(f1,h).(n+1).x + r2* fdif(f2,h).(n+1).x by Th7;
  hence thesis;
end;
