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
  (EqR1 "\/" EqR2) "\/" EqR3 = EqR1 "\/" (EqR2 "\/" EqR3)
proof
  for EqR4 be Equivalence_Relation of X holds EqR4 = EqR1 "\/" (EqR2 "\/"
  EqR3) implies (EqR1 "\/" EqR2) "\/" EqR3 c= EqR4
  proof
    let EqR4 be Equivalence_Relation of X;
A1: EqR2 \/ EqR3 c= EqR2 "\/" EqR3 by Def2;
    assume EqR4 = EqR1 "\/" (EqR2 "\/" EqR3);
    then
A2: EqR1 \/ (EqR2 "\/" EqR3) c= EqR4 by Def2;
    EqR2 "\/" EqR3 c= EqR1 \/ (EqR2 "\/" EqR3) by XBOOLE_1:7;
    then EqR2 "\/" EqR3 c= EqR4 by A2;
    then
A3: EqR2 \/ EqR3 c= EqR4 by A1;
    EqR2 c= EqR2 \/ EqR3 by XBOOLE_1:7;
    then
A4: EqR2 c= EqR4 by A3;
    EqR1 c= EqR1 \/ (EqR2 "\/" EqR3) by XBOOLE_1:7;
    then EqR1 c= EqR4 by A2;
    then EqR1 \/ EqR2 c= EqR4 by A4,XBOOLE_1:8;
    then
A5: EqR1 "\/" EqR2 c= EqR4 by Def2;
    EqR3 c= EqR2 \/ EqR3 by XBOOLE_1:7;
    then EqR3 c= EqR4 by A3;
    then (EqR1 "\/" EqR2) \/ EqR3 c= EqR4 by A5,XBOOLE_1:8;
    hence thesis by Def2;
  end;
  then
A6: (EqR1 "\/" EqR2) "\/" EqR3 c= EqR1 "\/" (EqR2 "\/" EqR3);
  for EqR4 be Equivalence_Relation of X holds EqR4 = (EqR1 "\/" EqR2) "\/"
  EqR3 implies EqR1 "\/" (EqR2 "\/" EqR3) c= EqR4
  proof
    let EqR4 be Equivalence_Relation of X;
A7: EqR1 \/ EqR2 c= EqR1 "\/" EqR2 by Def2;
    assume EqR4 = (EqR1 "\/" EqR2) "\/" EqR3;
    then
A8: (EqR1 "\/" EqR2) \/ EqR3 c= EqR4 by Def2;
    EqR1 "\/" EqR2 c= (EqR1 "\/" EqR2) \/ EqR3 by XBOOLE_1:7;
    then EqR1 "\/" EqR2 c= EqR4 by A8;
    then
A9: EqR1 \/ EqR2 c= EqR4 by A7;
    EqR3 c= (EqR1 "\/" EqR2) \/ EqR3 by XBOOLE_1:7;
    then
A10: EqR3 c= EqR4 by A8;
    EqR2 c= EqR1 \/ EqR2 by XBOOLE_1:7;
    then EqR2 c= EqR4 by A9;
    then EqR2 \/ EqR3 c= EqR4 by A10,XBOOLE_1:8;
    then
A11: EqR2 "\/" EqR3 c= EqR4 by Def2;
    EqR1 c= EqR1 \/ EqR2 by XBOOLE_1:7;
    then EqR1 c= EqR4 by A9;
    then EqR1 \/ (EqR2 "\/" EqR3) c= EqR4 by A11,XBOOLE_1:8;
    hence thesis by Def2;
  end;
  then EqR1 "\/" (EqR2 "\/" EqR3) c= (EqR1 "\/" EqR2) "\/" EqR3;
  hence thesis by A6;
end;
