theorem
  s,kai|= EG H iff ex pai being inf_path of R st pai.0 = s & for n being
  Element of NAT holds (pai.n),kai|= H
proof
A1: (ex pai being inf_path of R st pai.0 = s & for n being Element of NAT
holds (pai.n)|= Evaluate(H,kai) ) implies ex pai being inf_path of R st pai.0 =
  s & for n being Element of NAT holds (pai.n),kai|= H
  proof
    given pai be inf_path of R such that
A2: pai.0 = s and
A3: for n being Element of NAT holds (pai.n)|= Evaluate(H,kai);
    take pai;
    for n being Element of NAT holds (pai.n),kai|= H
    by A3;
    hence thesis by A2;
  end;
A4: (ex pai being inf_path of R st pai.0 = s & for n being Element of NAT
holds (pai.n),kai|= H ) implies ex pai being inf_path of R st pai.0 = s & for n
  being Element of NAT holds (pai.n)|= Evaluate(H,kai)
  proof
    given pai be inf_path of R such that
A5: pai.0 = s and
A6: for n being Element of NAT holds (pai.n),kai|= H;
    take pai;
    for n being Element of NAT holds (pai.n)|= Evaluate(H,kai)
    by A6,Def60;
    hence thesis by A5;
  end;
  s,kai|= EG H iff s|= EG Evaluate(H,kai) by Th8;
  hence thesis by A1,A4,Th15;
end;
