 reserve a,b,c,x for Real;
 reserve C for non empty set;

theorem
  for f,g being PartFunc of REAL, REAL st
  f is continuous & g is continuous &
  ex x being object st dom f /\ dom g = {x} &
  for x being object st x in dom f /\ dom g holds f.x = g.x holds
    ex h being PartFunc of REAL, REAL st
    h = f +* g & for x being Real st
    x in dom f /\ dom g holds h is_continuous_in x
  proof
    let f,g be PartFunc of REAL, REAL;
    assume
A1: f is continuous & g is continuous &
    ex x being object st dom f /\ dom g = {x} &
    for x being object st x in dom f /\ dom g holds f.x = g.x;
    reconsider h = f +* g as PartFunc of REAL, REAL;
    take h;
    thus h = f +* g;
    let x be Real;
J2: h | dom f = (g +* f) | dom f by FUNCT_4:34,PARTFUN1:def 4,A1
             .= f;
    assume x in dom f /\ dom g; then
ZR: x in dom f & x in dom g by XBOOLE_0:def 4; then
JJ: x in dom h by FUNCT_4:12;
    for r being Real st 0 < r
     ex s being Real st 0 < s &
    for x1 being Real st x1 in dom h & |.x1-x.|<s holds
      |.h.x1-h.x.| < r
    proof
      let r be Real;
      set hf = h | dom f;
      set hg = h | dom g;
SF:   x in dom hf by RELAT_1:57,ZR,JJ;
Sf:   x in dom hg by ZR;
      assume
R0:   0 < r;
      consider s2 being Real such that
SB:   0 < s2 & for x1 being Real st x1 in dom hf & |.x1-x.| < s2 holds
        |.hf.x1 - hf.x.| < r / 2 by J2,ZR,A1,FCONT_1:3,R0;
      consider s1 being Real such that
Sb:   0 < s1 & for x1 being Real st x1 in dom hg & |.x1-x.| < s1 holds
        |.hg.x1 - hg.x.| < r / 2 by FCONT_1:3,ZR,A1,R0;
      take s = min (s2,s1);
      thus 0 < s by SB,Sb,XXREAL_0:15;
SS:   r / 2 < r by XREAL_1:216,R0;
        let x1 be Real;
        assume
SC:     x1 in dom h & |.x1-x.| < s; then
        per cases by FUNCT_4:12;
        suppose ZT: x1 in dom f & x1 in dom g;
i1:       x1 in dom h by FUNCT_4:12,ZT; then
I1:       x1 in dom hf by ZT,RELAT_1:57;
          s <= s2 by XXREAL_0:17; then
          |.x1-x.| < s2 by XXREAL_0:2,SC; then
SD:       |.hf.x1-hf.x.| < r/2 by SB,i1,ZT,RELAT_1:57;
s1:       hf.x1 = h.x1 by FUNCT_1:47,I1;
          hf.x = h.x by FUNCT_1:47,SF;
          hence thesis by SS,XXREAL_0:2,SD,s1;
        end;
        suppose ZT: x1 in dom f & not x1 in dom g; then
i1:       x1 in dom h by FUNCT_4:12; then
I1:       x1 in dom hf by ZT,RELAT_1:57;
          s <= s2 by XXREAL_0:17; then
          |.x1-x.| < s2 by XXREAL_0:2,SC; then
SD:       |.hf.x1-hf.x.| < r/2 by SB,i1,ZT,RELAT_1:57;
s1:       hf.x1 = h.x1 by FUNCT_1:47,I1;
          hf.x = h.x by FUNCT_1:47,SF;
          hence thesis by SS,XXREAL_0:2,SD,s1;
        end;
        suppose
P1:       x1 in dom g;
          x1 in dom hg by P1; then
P2:       h.x1 = hg.x1 by FUNCT_1:47;
P3:       h.x = hg.x by FUNCT_1:47,Sf;
          s <= s1 by XXREAL_0:17; then
          |.x1-x.| < s1 by XXREAL_0:2,SC; then
          |.hg.x1-hg.x.| < r/2 by Sb,P1;
          hence thesis by SS,XXREAL_0:2,P3,P2;
      end;
    end;
    hence thesis by FCONT_1:3;
  end;
