theorem Th55:
  p=>q in F implies F\/{p}|-'G' q
 proof
  p in {p} by TARSKI:def 1;
  then p in F\/{p} by XBOOLE_0:def 3;
  then A1: F\/{p}|-p by Th42;
  assume p=>q in F;
  then p=>q in F\/{p} by XBOOLE_0:def 3;
  then F\/{p}|-p=>q by Th42;
  then F\/{p}|-q by A1,Th43;
  hence F\/{p}|-'G' q by Th54;
 end;
