
theorem
  for FT1,FT2 being non empty RelStr,A being Subset of FT1, B being
Subset of FT2, f being Function of FT1, FT2 st A is connected & f is_continuous
  0 & B=f.:A holds B is connected
proof
  let FT1,FT2 be non empty RelStr,A be Subset of FT1, B be Subset of FT2, f be
  Function of FT1, FT2;
  assume that
A1: A is connected and
A2: f is_continuous 0 and
A3: B=f.:A;
  for B2,C2 being Subset of FT2 st B = B2 \/ C2 & B2 <> {} & C2 <> {} & B2
  misses C2 holds B2^b meets C2
  proof
    let B2,C2 be Subset of FT2;
    assume that
A4: B = B2 \/ C2 and
A5: B2 <> {} and
A6: C2 <> {} and
A7: B2 misses C2;
    reconsider C1=f"C2 as Subset of FT1;
    reconsider C10=C1 /\ A as Subset of FT1;
    reconsider B1 = f"B2 as Subset of FT1;
    reconsider B10=B1/\A as Subset of FT1;
A8: C10 c= C1 by XBOOLE_1:17;
    set x6 = the Element of C2;
    x6 in B by A4,A6,XBOOLE_0:def 3;
    then consider z6 being object such that
A9: z6 in dom f and
A10: z6 in A and
A11: x6=f.z6 by A3,FUNCT_1:def 6;
    z6 in f"C2 by A6,A9,A11,FUNCT_1:def 7;
    then
A12: C10<>{} by A10,XBOOLE_0:def 4;
    set x5 = the Element of B2;
    x5 in B by A4,A5,XBOOLE_0:def 3;
    then consider z5 being object such that
A13: z5 in dom f and
A14: z5 in A and
A15: x5=f.z5 by A3,FUNCT_1:def 6;
    A c= the carrier of FT1;
    then
A16: A c= dom f by FUNCT_2:def 1;
    B2 /\ C2 = {} by A7,XBOOLE_0:def 7;
    then f"(B2 /\ C2) = {};
    then B10 c= B1 & (f"B2) /\ (f"C2) = {} by FUNCT_1:68,XBOOLE_1:17;
    then B10 /\ C10= {} by A8,XBOOLE_1:3,27;
    then
A17: B10 misses C10 by XBOOLE_0:def 7;
    (f"B2) \/ (f"C2)=f"(f.:A) by A3,A4,RELAT_1:140;
    then A c= B1 \/ C1 by A16,FUNCT_1:76;
    then A c= A/\(B1\/C1) by XBOOLE_1:19;
    then
A18: A c= B10 \/ C10 by XBOOLE_1:23;
    B10 c= A & C10 c= A by XBOOLE_1:17;
    then B10 \/ C10 c= A by XBOOLE_1:8;
    then
A19: A=B10 \/ C10 by A18,XBOOLE_0:def 10;
    z5 in f"B2 by A5,A13,A15,FUNCT_1:def 7;
    then B10<>{} by A14,XBOOLE_0:def 4;
    then B10^b meets C10 by A1,A19,A12,A17;
    then
A20: B10^b /\ C10 <> {} by XBOOLE_0:def 7;
    reconsider B20 = f.:B1 as Subset of FT2;
A21: dom f=the carrier of FT1 by FUNCT_2:def 1;
    f.:B1 c= B2
    proof
      let y be object;
      assume y in f.:B1;
      then ex x2 being object st x2 in dom f & x2 in B1 & y=f.x2
by FUNCT_1:def 6;
      hence thesis by FUNCT_1:def 7;
    end;
    then
A22: B20^b c= B2^b by FIN_TOPO:14;
    f.:(B1^b) c= B20^b by A2,Th16;
    then f.:(B1^b) c= B2^b by A22;
    then
A23: (f.:(B1^b)) /\ (f.:C1) c= (f.:(B1^b)) /\ C2 & (f.:(B1^b)) /\ C2 c= B2
    ^b /\ C2 by FUNCT_1:75,XBOOLE_1:26;
    B10^b c= B1^b by FIN_TOPO:14,XBOOLE_1:17;
    then B1^b /\ C1 <> {} by A8,A20,XBOOLE_1:3,27;
    then f.:(B1^b /\ C1) <> {} by A21,RELAT_1:119;
    then (f.:(B1^b)) /\ (f.:C1) <> {} by RELAT_1:121,XBOOLE_1:3;
    then B2^b /\ C2 <>{} by A23;
    hence thesis by XBOOLE_0:def 7;
  end;
  hence thesis;
end;
