theorem Th53:
  for G being Graph, c1, c2 be Element of G-CycleSet st G-VSet rng
  c1 meets G-VSet rng c2 & rng c1 misses rng c2 & (c1 <> {} or c2 <> {}) holds
  CatCycles(c1, c2) is non empty
proof
  let G be Graph, c1, c2 be Element of G-CycleSet;
  assume that
A1: G-VSet rng c1 meets G-VSet rng c2 and
A2: rng c1 misses rng c2 and
A3: c1 <> {} or c2 <> {};
  consider v being Vertex of G such that
A4: v = the Element of (G-VSet rng c1) /\ (G-VSet rng c2) and
A5: CatCycles(c1, c2) = Rotate(c1, v)^Rotate(c2, v) by A1,A2,Def10;
A6: (G-VSet rng c1) /\ (G-VSet rng c2) <> {} by A1;
  then
A7: v in (G-VSet rng c1) by A4,XBOOLE_0:def 4;
A8: v in (G-VSet rng c2) by A4,A6,XBOOLE_0:def 4;
  per cases by A3;
  suppose
    c1 <> {};
    then rng Rotate(c1, v) <> {} by A7,Lm5;
    hence thesis by A5,FINSEQ_1:35,RELAT_1:38;
  end;
  suppose
   c2 <> {};
    then rng Rotate(c2, v) <> {} by A8,Lm5;
    hence thesis by A5,FINSEQ_1:35,RELAT_1:38;
  end;
end;
