reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  for PA being a_partition of Y holds
    %I(Y) '\/' PA = PA & %I(Y) '/\' PA = %I(Y)
proof
  let PA be a_partition of Y;
A1: ERl(%I(Y)) = id Y by Th34;
  A2: ERl
(%I(Y) '\/' PA) = ERl(%I(Y)) "\/" ERl(PA) & ERl(%I(Y)) \/ ERl(PA) c= ERl(
  %I(Y)) "\/" ERl(PA) by Th23,EQREL_1:def 2;
A3: ERl(%I(Y)) \/ ERl(PA) = id Y \/ ERl(PA) by Th34;
 %I(Y) '<' PA by Th32;
then A4: ERl(%I(Y)) c= ERl(PA) by Th20;
 for z being object st z in id Y \/ ERl(PA) holds z in ERl PA
  proof
    let z be object;
    assume
A5: z in id Y \/ ERl(PA);
    now per cases by A5,XBOOLE_0:def 3;
      case z in id Y;
        hence thesis by A1,A4;
      end;
      case z in ERl(PA);
        hence thesis;
      end;
    end;
    hence thesis;
  end; then
A6: id Y \/ ERl(PA) c= ERl(PA);
   ERl(PA) c= id Y \/ ERl(PA) by XBOOLE_1:7;
   then id Y \/ ERl(PA) = ERl(PA) by A6,XBOOLE_0:def 10; then
A7: PA '<' %I(Y) '\/' PA by A2,A3,Th20;
   %I(Y) '<' PA by Th32;
   then %I(Y) '\/' PA '<' PA by Th29;
   hence %I(Y) '\/' PA = PA by A7,Th4;
   ERl(%I(Y) '/\' PA) = ERl(%I(Y)) /\ ERl(PA) by Th24
    .= id Y /\ ERl(PA) by Th34
    .= id Y by EQREL_1:10
    .= ERl(%I(Y)) by Th34;
  hence %I(Y) '/\' PA = %I(Y) by Th25;
end;
