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 Th32:
  [!(f1+f2),x0,x1!] = [!f1,x0,x1!] + [!f2,x0,x1!]
proof
 reconsider xx0=x0, xx1=x1 as Element of REAL by XREAL_0:def 1;
  [!(f1+f2),x0,x1!] = (f1.xx0+f2.xx0-(f1+f2).xx1)/(xx0-xx1) by VALUED_1:1
    .= (f1.x0+f2.x0-(f1.x1+f2.x1))/(x0-x1) by VALUED_1:1
    .= ((f1.x0-f1.x1)+(f2.x0-f2.x1))/(x0-x1)
    .= [!f1,x0,x1!]+[!f2,x0,x1!] by XCMPLX_1:62;
  hence thesis;
end;
