
theorem Th6:
  for f being Function of R^1, R^1, x being Point of R^1 for g
being PartFunc of REAL, REAL, x1 being Real
  st f is_continuous_at x & f = g & x = x1 holds g is_continuous_in x1
proof
  let f be Function of R^1, R^1, x be Point of R^1;
  let g be PartFunc of REAL, REAL, x1 be Real;
  assume that
A1: f is_continuous_at x and
A2: f = g and
A3: x = x1;
  for N1 being Neighbourhood of g.x1 ex N being Neighbourhood of x1 st g.:
  N c= N1
  proof
    reconsider fx = f.x as Point of R^1;
    let N1 be Neighbourhood of g.x1;
    reconsider N19 = N1 as Subset of R^1 by TOPMETR:17;
    reconsider N2 = N1 as Subset of RealSpace by TOPMETR:12,17,def 6;
    N2 in Family_open_set RealSpace by Lm3;
    then N2 in the topology of TopSpaceMetr RealSpace by TOPMETR:12;
    then N19 is open by TOPMETR:def 6;
    then reconsider G = N19 as a_neighborhood of fx by A2,A3,CONNSP_2:3
,RCOMP_1:16;
    consider H being a_neighborhood of x such that
A4: f.:H c= G by A1,TMAP_1:def 2;
    consider V being Subset of R^1 such that
A5: V is open and
A6: V c= H and
A7: x in V by CONNSP_2:6;
    reconsider V1 = V as Subset of REAL by TOPMETR:17;
    V in the topology of R^1 by A5;
    then V in Family_open_set RealSpace by TOPMETR:12,def 6;
    then V1 is open by Lm4;
    then consider N being Neighbourhood of x1 such that
A8: N c= V1 by A3,A7,RCOMP_1:18;
    N c= H by A6,A8;
    then g.:N c= g.:H by RELAT_1:123;
    hence thesis by A2,A4,XBOOLE_1:1;
  end;
  hence thesis by FCONT_1:5;
end;
