theorem Th35:
  a,b '||' a,c & a,b '||' a,d implies a,b '||' c,d
proof
  assume that
A1: a,b '||' a,c and
A2: a,b '||' a,d;
  now
    assume that
A3: a<>b and
A4: a<>c;
    a,c '||' a,d by A1,A2,A3,Def11;
    then a,c '||' c,d by Th24;
    hence thesis by A1,A4,Th26;
  end;
  hence thesis by A2,Th20;
end;
