theorem Th41:
  b in B implies ex c st c c= b & c in mi B
proof
  assume
A1: b in B;
  defpred P[Element of DISJOINT_PAIRS A,Element of DISJOINT_PAIRS A] means $1
  c= $2;
A2: for a holds P[a,a];
A3: for a,b,c st P[a,b] & P[b,c] holds P[a,c] by Th2;
  for a st a in B ex b st b in B & P[b,a] & for c st c in B & P[c,b] holds
  P[b,c] from FRAENKEL:sch 23(A2,A3);
  then consider c such that
A4: c in B and
A5: c c= b and
A6: for a st a in B & a c= c holds c c= a by A1;
  take c;
  thus c c= b by A5;
  now
    let b;
    assume that
A7: b in B and
A8: b c= c;
    c c= b by A6,A7,A8;
    hence b = c by A8,Th1;
  end;
  hence thesis by A4,Th39;
end;
