reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem Th29:
  for E being Equivalence_Relation of X st E = EqR1 \/ EqR2 holds
for x st x in X holds Class(E,x) = Class(EqR1,x) or Class(E,x) = Class(EqR2,x)
proof
  let E be Equivalence_Relation of X such that
A1: E = EqR1 \/ EqR2;
  for x st x in X holds Class(E,x) = Class(EqR1,x) or Class(E,x) = Class(
  EqR2,x)
  proof
    let x such that
    x in X;
    assume that
A2: not Class(E,x) = Class(EqR1,x) and
A3: not Class(E,x) = Class(EqR2,x);
    consider y being object such that
A4: y in Class(E,x) & not y in Class(EqR1,x) or y in Class(EqR1,x) &
    not y in Class(E,x) by A2,TARSKI:2;
A5: now
      assume that
A6:   y in Class(EqR1,x) and
A7:   not y in Class(E,x);
      [y,x] in EqR1 by A6,Th19;
      then [y,x] in E by A1,XBOOLE_0:def 3;
      hence contradiction by A7,Th19;
    end;
    then
A8: [y,x] in E by A4,Th19;
    consider z being object such that
A9: z in Class(E,x) & not z in Class(EqR2,x) or z in Class(EqR2,x) &
    not z in Class(E,x) by A3,TARSKI:2;
A10: now
      assume that
A11:  z in Class(EqR2,x) and
A12:  not z in Class(E,x);
      [z,x] in EqR2 by A11,Th19;
      then [z,x] in E by A1,XBOOLE_0:def 3;
      hence contradiction by A12,Th19;
    end;
    then
A13: [z,x] in E by A9,Th19;
    not [z,x] in EqR2 by A9,A10,Th19;
    then
A14: [z,x] in EqR1 by A1,A13,XBOOLE_0:def 3;
A15: now
      assume [y,z] in EqR1;
      then
A16:  [z,y] in EqR1 by Th6;
      [x,z] in EqR1 by A14,Th6;
      then [x,y] in EqR1 by A16,Th7;
      then [y,x] in EqR1 by Th6;
      hence contradiction by A4,A5,Th19;
    end;
    not [y,x] in EqR1 by A4,A5,Th19;
    then
A17: [y,x] in EqR2 by A1,A8,XBOOLE_0:def 3;
A18: now
      assume
A19:  [y,z] in EqR2;
      [x,y] in EqR2 by A17,Th6;
      then [x,z] in EqR2 by A19,Th7;
      then [z,x] in EqR2 by Th6;
      hence contradiction by A9,A10,Th19;
    end;
    [x,z] in E by A13,Th6;
    then [y,z] in E by A8,Th7;
    hence contradiction by A1,A18,A15,XBOOLE_0:def 3;
  end;
  hence thesis;
end;
