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 Th65:
  congr r,s,a,b & congr r,s,c,d implies congr a,b,c,d
proof
  assume that
A1: congr r,s,a,b and
A2: congr r,s,c,d;
A3: now
    assume that
    r<>s and
A4: ( not r,s,a are_collinear)& not r,s,c are_collinear;
    parallelogram r,s,a,b & parallelogram r,s,c,d by A1,A2,A4,Th59;
    hence thesis;
  end;
A5: now
    assume that
A6: r<>s and
A7: r,s,c are_collinear;
    consider p,q such that
A8: parallelogram p,q,r,s and
A9: parallelogram p, q,a,b by A1,A6;
    parallelogram p,q,c,d by A2,A7,A8,Th61;
    hence thesis by A9;
  end;
A10: now
    assume that
A11: r<>s and
A12: r,s,a are_collinear;
    consider p,q such that
A13: parallelogram p,q,r,s and
A14: parallelogram p,q,c,d by A2,A11;
    parallelogram p,q,a,b by A1,A12,A13,Th61;
    hence thesis by A14;
  end;
  r=s implies thesis by A1,A2,Th54;
  hence thesis by A10,A5,A3;
end;
