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 Th63:
  ex d st congr a,b,c,d
proof
A1: now
    assume a=b;
    then congr a,b,c,c;
    hence thesis;
  end;
A2: now
    assume that
A3: a<>b and
A4: a,b,c are_collinear;
    consider p,q such that
A5: parallelogram a,b,p,q by A3,Lm1;
    not p,q,c are_collinear by A4,A5,Th39;
    then consider d such that
A6: parallelogram p,q,c,d by Th44;
    parallelogram p,q,a,b by A5,Th43;
    then congr a,b,c,d by A6;
    hence thesis;
  end;
  now
    assume that
    a<>b and
A7: not a,b,c are_collinear;
    ex d st parallelogram a,b,c,d by A7,Th44;
    hence thesis by Th60;
  end;
  hence thesis by A1,A2;
end;
