reserve X,Y for set;
reserve R for domRing-like commutative Ring;
reserve c for Element of R;

theorem Th20:
  for R being associative non empty multLoopStr
  for a,b being Element of R holds
    Class a meets Class b implies Class a = Class b
proof
  let R be associative non empty multLoopStr;
  let a,b be Element of R;
  assume Class a /\ Class b <> {};
  then Class a meets Class b;
  then consider Z being object such that
A1: Z in Class a and
A2: Z in Class b by XBOOLE_0:3;
  reconsider Z as Element of R by A1;
A3: Z is_associated_to b by A2,Def5;
A4: Z is_associated_to a by A1,Def5;
A5: for c being Element of R holds c in Class b implies c in Class a
  proof
    let c be Element of R;
    assume c in Class b;
    then c is_associated_to b by Def5;
    then Z is_associated_to c by A3,Th4;
    then a is_associated_to c by A4,Th4;
    hence thesis by Def5;
  end;
  for c be Element of R holds c in Class a implies c in Class b
  proof
    let c be Element of R;
    assume c in Class a;
    then c is_associated_to a by Def5;
    then Z is_associated_to c by A4,Th4;
    then b is_associated_to c by A3,Th4;
    hence thesis by Def5;
  end;
  hence thesis by A5,SUBSET_1:3;
end;
