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 Th60:
  parallelogram a,b,c,d implies congr a,b,c,d
proof
A1: a,b // a,b by Th1;
  assume
A2: parallelogram a,b,c,d;
  then
A3: ( not a,c,b are_collinear)& a<>b by Th36,Th38;
  a<>c by A2,Th36;
  then consider p such that
A4: a,c,p are_collinear and
A5: a<>p and
A6: c <>p by Th48;
  a,c // a,p by A4;
  then consider q such that
A7: parallelogram a,p,b,q by A5,A1,A3,Th23,Th44;
  parallelogram a,b,p,q by A7,Th43;
  then parallelogram c,d,p,q by A2,A4,A6,Th51;
  then
A8: parallelogram p,q,c,d by Th43;
  parallelogram p,q,a,b by A7,Th43;
  hence thesis by A8;
end;
