reserve A,B,p,q,r for Element of LTLB_WFF,
  M for LTLModel,
  j,k,n for Element of NAT,
  i for Nat,
  X for Subset of LTLB_WFF,
  F for finite Subset of LTLB_WFF,
  f for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y,z for set,
  P,Q,R for PNPair;

theorem Th19: Q in compn P & R in untn P implies Q is_completion_of R
  proof
    assume that
A1: Q in compn P and
A2: R in untn P;
    consider Q1 be PNPair such that
A3: Q1 = Q and
A4: Q1 in comp untn P by A1;
A5: ex R1 be PNPair st R1 = R & rng R1`1 = untn rng P`1 &
    rng R1 `2 = untn rng P`2 by A2;
    consider x such that
A6: Q1 in x and
A7: x in {comp S where S is PNPair:S in untn P} by A4,TARSKI:def 4;
    consider S be PNPair such that
A8: x = comp S and
A9: S in untn P by A7;
    consider Q2 be consistent PNPair such that
A10: Q2 = Q1 and
A11: Q2 is_completion_of S by A6,A8;
A12: ex S1 be PNPair st S1 = S & rng S1`1 = untn rng P`1 &
     rng S1`2 = untn rng P`2 by A9;
     tau rng S = rng Q2 by A11;
     then A13: tau rng R = rng Q by A3,A10,A12,A5;
     rng R`1 c= rng Q`1 & rng R`2 c= rng Q`2 by A11,A3,A10,A12, A5;
     hence Q is_completion_of R by A13;
   end;
