reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
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
  set g1=r1(#)f1;
  set g2=r2(#)f2;
  [!(r1(#)f1-r2(#)f2),x0,x1!] = [!g1,x0,x1!] - [!g2,x0,x1!] by Th25
    .= r1* [!f1,x0,x1!] - [!g2,x0,x1!] by DIFF_1:31
    .= r1* [!f1,x0,x1!] - r2* [!f2,x0,x1!] by DIFF_1:31;
  hence thesis;
end;
