reserve I,X,x,d,i for set;
reserve M for ManySortedSet of I;
reserve EqR1,EqR2 for Equivalence_Relation of X;
reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve EqR,EqR1,EqR2,EqR3,EqR4 for Equivalence_Relation of M;

theorem Th8:
  (EqR1 "\/" EqR2) "\/" EqR3 = EqR1 "\/" (EqR2 "\/" EqR3)
proof
  for EqR4 holds EqR4 = EqR1 "\/" (EqR2 "\/" EqR3) implies (EqR1 "\/"
  EqR2) "\/" EqR3 c= EqR4
  proof
    let EqR4;
A1: EqR2 (\/) EqR3 c= EqR2 "\/" EqR3 by Th4;
    assume EqR4 = EqR1 "\/" (EqR2 "\/" EqR3);
    then
A2: EqR1 (\/) (EqR2 "\/" EqR3) c= EqR4 by Th4;
    EqR2 "\/" EqR3 c= EqR1 (\/) (EqR2 "\/" EqR3) by PBOOLE:14;
    then EqR2 "\/" EqR3 c= EqR4 by A2,PBOOLE:13;
    then
A3: EqR2 (\/) EqR3 c= EqR4 by A1,PBOOLE:13;
    EqR2 c= EqR2 (\/) EqR3 by PBOOLE:14;
    then
A4: EqR2 c= EqR4 by A3,PBOOLE:13;
    EqR1 c= EqR1 (\/) (EqR2 "\/" EqR3) by PBOOLE:14;
    then EqR1 c= EqR4 by A2,PBOOLE:13;
    then EqR1 (\/) EqR2 c= EqR4 by A4,PBOOLE:16;
    then
A5: EqR1 "\/" EqR2 c= EqR4 by Th5;
    EqR3 c= EqR2 (\/) EqR3 by PBOOLE:14;
    then EqR3 c= EqR4 by A3,PBOOLE:13;
    then (EqR1 "\/" EqR2) (\/) EqR3 c= EqR4 by A5,PBOOLE:16;
    hence thesis by Th5;
  end;
  then
A6: (EqR1 "\/" EqR2) "\/" EqR3 c= EqR1 "\/" (EqR2 "\/" EqR3);
  for EqR4 holds EqR4 = (EqR1 "\/" EqR2) "\/" EqR3 implies EqR1 "\/" (EqR2
  "\/" EqR3) c= EqR4
  proof
    let EqR4;
A7: EqR1 (\/) EqR2 c= EqR1 "\/" EqR2 by Th4;
    assume EqR4 = (EqR1 "\/" EqR2) "\/" EqR3;
    then
A8: (EqR1 "\/" EqR2) (\/) EqR3 c= EqR4 by Th4;
    EqR1 "\/" EqR2 c= (EqR1 "\/" EqR2) (\/) EqR3 by PBOOLE:14;
    then EqR1 "\/" EqR2 c= EqR4 by A8,PBOOLE:13;
    then
A9: EqR1 (\/) EqR2 c= EqR4 by A7,PBOOLE:13;
    EqR3 c= (EqR1 "\/" EqR2) (\/) EqR3 by PBOOLE:14;
    then
A10: EqR3 c= EqR4 by A8,PBOOLE:13;
    EqR2 c= EqR1 (\/) EqR2 by PBOOLE:14;
    then EqR2 c= EqR4 by A9,PBOOLE:13;
    then EqR2 (\/) EqR3 c= EqR4 by A10,PBOOLE:16;
    then
A11: EqR2 "\/" EqR3 c= EqR4 by Th5;
    EqR1 c= EqR1 (\/) EqR2 by PBOOLE:14;
    then EqR1 c= EqR4 by A9,PBOOLE:13;
    then EqR1 (\/) (EqR2 "\/" EqR3) c= EqR4 by A11,PBOOLE:16;
    hence thesis by Th5;
  end;
  then EqR1 "\/" (EqR2 "\/" EqR3) c= (EqR1 "\/" EqR2) "\/" EqR3;
  hence thesis by A6,PBOOLE:146;
end;
