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;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;

theorem Th41:
  for S1 being a_partition of X, x,y being Element of X holds
  EqClass(x,S1) meets EqClass(y,S1) implies EqClass(x,S1) = EqClass(y,S1)
proof
  let S1 be a_partition of X;
  let x,y be Element of X;
  consider EQR being Equivalence_Relation of X such that
A1: S1 = Class EQR by Th34;
A2: y in Class(EQR,y) by Th20;
  Class(EQR,y) in S1 by A1,Def3;
  then
A3: Class(EQR,y) = EqClass(y,S1) by A2,Def6;
A4: x in Class(EQR,x) by Th20;
  Class(EQR,x) in S1 by A1,Def3;
  then
A5: Class(EQR,x) = EqClass(x,S1) by A4,Def6;
  assume EqClass(x,S1) meets EqClass(y,S1);
  hence thesis by A5,A3,Th24;
end;
