reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th57:
  congr a,b,c,d implies a,b // c,d
proof
  assume
A1: congr a,b,c,d;
  now
    assume a<>b;
    then consider p,q such that
A2: parallelogram p,q,a,b and
A3: parallelogram p, q,c,d by A1;
A4: p,q // c,d by A3;
    p<>q & p,q // a,b by A2,Th36;
    hence thesis by A4,Def1;
  end;
  hence thesis by Th3;
end;
