reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;
reserve T for non empty TopSpace,
  S for TopSpace,
  P1 for Subset of S,
  f for Function of T, S;
reserve T for TopSpace,
  S for non empty TopSpace,
  P for Subset of T,
  f for Function of T, S;

theorem Th61: :: TOPREAL5:5, AK, 21.02.2006
  for X,Y being non empty TopSpace for f being Function of X,Y, A
  being Subset of X st f is continuous & A is connected holds f.:A is connected
proof
  let X,Y be non empty TopSpace;
  let f be Function of X,Y, A be Subset of X;
  assume
A1: f is continuous;
  assume
A2: A is connected;
  assume not f.:A is connected;
  then consider P,Q being Subset of Y such that
A3: f.:A = P \/ Q and
A4: P,Q are_separated and
A5: P <> {}(Y) and
A6: Q <> {}(Y) by CONNSP_1:15;
  reconsider P1=f"P,Q1=f"Q as Subset of X;
  set P2=P1/\A,Q2=Q1/\A;
  set y = the Element of P;
  y in f.:A by A3,A5,XBOOLE_0:def 3;
  then consider x being object such that
A7: x in dom f and
A8: x in A and
A9: y=f.x by FUNCT_1:def 6;
  x in f"P by A5,A7,A9,FUNCT_1:def 7;
  then
A10: P2<>{} by A8,XBOOLE_0:def 4;
A11: the carrier of X=dom f by FUNCT_2:def 1;
  P misses Cl Q by A4,CONNSP_1:def 1;
  then
A12: f"(P) /\ f"(Cl Q) = f"(P /\ Cl Q) & f"(P /\ Cl Q)=f"({}Y) by FUNCT_1:68
,XBOOLE_0:def 7;
  Cl Q1 c=f"(Cl Q) by A1,Th44;
  then P1 /\ Cl Q1 = {} by A12,XBOOLE_1:3,26;
  then
A13: P1 misses Cl Q1 by XBOOLE_0:def 7;
  Cl P misses Q by A4,CONNSP_1:def 1;
  then
A14: f"(Cl P) /\ f"Q = f"(Cl P /\ Q) & f"(Cl P /\ Q)=f"({}Y) by FUNCT_1:68
,XBOOLE_0:def 7;
  Cl P1 c=f"(Cl P) by A1,Th44;
  then Cl P1 /\ Q1 = {}X by A14,XBOOLE_1:3,26;
  then Cl P1 misses Q1 by XBOOLE_0:def 7;
  then
A15: P1,Q1 are_separated by A13,CONNSP_1:def 1;
  set z = the Element of Q;
  z in P \/ Q by A6,XBOOLE_0:def 3;
  then consider x1 being object such that
A16: x1 in dom f and
A17: x1 in A and
A18: z=f.x1 by A3,FUNCT_1:def 6;
  x1 in f"Q by A6,A16,A18,FUNCT_1:def 7;
  then
A19: Q2<>{} by A17,XBOOLE_0:def 4;
  f"(f.:A)=f"P \/ f"Q by A3,RELAT_1:140;
  then
A20: A=(P1 \/ Q1)/\A by A11,FUNCT_1:76,XBOOLE_1:28
    .=P2 \/ Q2 by XBOOLE_1:23;
  P2 c=P1 & Q2 c=Q1 by XBOOLE_1:17;
  then ex P3,Q3 being Subset of X st A = P3 \/ Q3 & P3,Q3 are_separated & P3
  <> {}(X) & Q3 <> {}(X) by A15,A20,A10,A19,CONNSP_1:7;
  hence contradiction by A2,CONNSP_1:15;
end;
