theorem Th38:
  a in A & b in A & c in C & d in C & A // C implies a,b // c,d
proof
  assume that
A1: a in A and
A2: b in A and
A3: c in C and
A4: d in C and
A5: A // C;
  now
A6: C is being_line by A5,Th35;
    assume that
A7: a<>b and
A8: c <>d;
    A is being_line by A5;
    hence thesis by A1,A2,A3,A4,A5,A7,A8,A6,Th37;
  end;
  hence thesis by Th2;
end;
