
theorem Th4:
  for X being non empty TopSpace
  for f,g being continuous Function of the carrier of X,COMPLEX
   holds f+g is continuous Function of the carrier of X,COMPLEX
proof
  let X be non empty TopSpace,
  f,g be continuous Function of the carrier of X,COMPLEX;
  set h=f+g;
A1: rng h c= COMPLEX;
  dom h = dom f /\ dom g by VALUED_1:def 1
     .= (the carrier of X) /\ dom g by PARTFUN1:def 2
     .= (the carrier of X) /\ the carrier of X by PARTFUN1:def 2;
  then reconsider h as Function of the carrier of X,COMPLEX by A1,FUNCT_2:2;
A2:for x be Point of X holds h.x=f.x + g.x by VALUED_1:1;
  for p being Point of X,V being Subset of COMPLEX
    st h.p in V & V is open holds
      ex W being Subset of X st p in W & W is open & h.:W c= V
  proof
    let p be Point of X,V be Subset of COMPLEX;
    assume
A3:   h.p in V & V is open;
    reconsider z0=h.p as Complex;
    consider N being Neighbourhood of z0 such that
A4:   N c= V by A3,CFDIFF_1:9;
    consider r being Real such that
A5: 0<r & {y where y is Complex:|.(y-z0).| < r } c= N by CFDIFF_1:def 5;
    set S={y where y is Complex:|.(y-z0).| < r };
    reconsider z1=f.p as Complex;
    set S1={y where y is Complex:|.(y-z1).| < r/2 };
    S1 c= COMPLEX
    proof
      let z be object;
      assume z in S1;
      then ex y being Complex st z = y & |.(y - z1).| < r/2;
      hence z in COMPLEX by XCMPLX_0:def 2;
    end; then
    reconsider T1=S1 as Subset of COMPLEX;
A6: T1 is open by CFDIFF_1:13;
    |.(z1 - z1).|=0;
    then z1 in S1 by A5;
    then consider W1 being Subset of X such that
A7:     p in W1 & W1 is open & f.:W1 c= S1 by A6,Th3;
    reconsider z2=g.p as Complex;
    set S2={y where y is Complex:|.(y-z2).| < r/2 };
    S2 c= COMPLEX
    proof
      let z be object;
      assume z in S2;
      then ex y being Complex st z = y & |.(y - z2).| < r/2;
      hence z in COMPLEX by XCMPLX_0:def 2;
    end;
    then reconsider T2=S2 as Subset of COMPLEX;
A8: T2 is open by CFDIFF_1:13;
    |.(z2 - z2).|=0;
    then z2 in S2 by A5;
    then consider W2 being Subset of X such that
A9: p in W2 & W2 is open & g.:W2 c= S2 by A8,Th3;
    set W=W1 /\ W2;
A10:W is open by A7,A9,TOPS_1:11;
A11:p in W by A7,A9,XBOOLE_0:def 4;
    h.:W c= S
    proof
        let z3 be object;
        assume z3 in h.:W;
        then consider x3 being object such that
A12:      x3 in dom h & x3 in W & h.x3=z3 by FUNCT_1:def 6;
A13:    x3 in W1 by A12,XBOOLE_0:def 4;
        reconsider px=x3 as Point of X by A12;
A14:    px in the carrier of X;
        then px in dom f by FUNCT_2:def 1;
        then f.px in f.:W1 by A13,FUNCT_1:def 6;
        then
A15:    f.px in S1 by A7;
        reconsider a1=f.px as Complex;
A16:    ex aa1 be Complex st f.px = aa1 & |.(aa1-z1).| < r/2 by A15;
A17:    x3 in W2 by A12,XBOOLE_0:def 4;
        px in dom g by A14,FUNCT_2:def 1;
        then g.px in g.:W2 by A17,FUNCT_1:def 6;
        then
A18:      g.px in S2 by A9;
        reconsider a2=g.px as Complex;
        ex aa2 be Complex st g.px = aa2 & |.(aa2-z2).| < r/2 by A18;
        then
A19:      |.(a2 - z2).| < r/2;
        |.(h.x3 - z0).| = |.((f.px+g.px) - z0).| by A2
                       .= |.((f.px+g.px) - (f.p+g.p)).| by A2
                       .= |.((a1 - z1) + (a2 - z2)).|;
        then
A20:      |.(h.px - z0).| <= |.(a1-z1).|+|.(a2-z2).| by COMPLEX1:56;
A21:    |.(a1-z1).|+|.(a2-z2).| < r/2 + |.(a2-z2).| by A16,XREAL_1:8;
        r/2 + |.(a2-z2).| < r/2 + r/2 by A19,XREAL_1:8;
        then |.(a1-z1).|+|.(a2-z2).| < r by A21,XXREAL_0:2;
        then |.(h.px - z0).| < r by A20,XXREAL_0:2;
        hence z3 in S by A12;
    end;
    then h.:W c= N by A5;
    hence ex W being Subset of X st p in W & W is open & h.:W c= V
                                        by A10,A11,A4,XBOOLE_1:1;
  end;
  hence thesis by Th3;
end;
