
theorem Th49: :: Mycn1:
 Mycielskian 0 = CompleteRelStr 2 &
 for k being Nat holds Mycielskian (k+1) = Mycielskian Mycielskian k
proof
  consider myc being Function such that
A1: Mycielskian 0 = myc.0 and
   dom myc = NAT and
A2: myc.0 = CompleteRelStr 2 and
   for k being Nat, R being NatRelStr of 3*2|^k-'1
       st R = myc.k holds myc.(k+1) = Mycielskian R by Def10;
 thus Mycielskian 0 = CompleteRelStr 2 by A1,A2;
 let k be Nat;
  consider myc1 being Function such that
A3: Mycielskian k = myc1.k and
A4: dom myc1 = NAT & myc1.0 = CompleteRelStr 2 &
    for k being Nat, R being NatRelStr of 3*2|^k-'1
       st R = myc1.k holds myc1.(k+1) = Mycielskian R by Def10;
  consider myc2 being Function such that
A5: Mycielskian (k+1) = myc2.(k+1) and
A6: dom myc2 = NAT & myc2.0 = CompleteRelStr 2 and
A7: for k being Nat, R being NatRelStr of 3*2|^k-'1
       st R = myc2.k holds myc2.(k+1) = Mycielskian R by Def10;
    myc1 = myc2 by A4,A6,A7,Lm1;
  hence thesis by A3,A7,A5;
end;
