
theorem Th1:
  for F being Function, i, xi being set, Ai being Subset of F.i
  holds proj(F,i)"({xi}) meets proj(F,i)"Ai implies xi in Ai
proof
  let F be Function, i be set, xi be set, Ai be Subset of F.i;
  set f = the Element of proj(F,i)"({xi}) /\ proj(F,i)"Ai;
  assume
A1: proj(F,i)"({xi}) /\ proj(F,i)"Ai <> {};
  then f in proj(F,i)"({xi}) by XBOOLE_0:def 4;
  then proj(F,i).f in {xi} by FUNCT_1:def 7;
  then
A2: proj(F,i).f = xi by TARSKI:def 1;
  f in proj(F,i)"Ai by A1,XBOOLE_0:def 4;
  hence thesis by A2,FUNCT_1:def 7;
end;
