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 Th62:
  congr a,b,c,x & congr a,b,c,y implies x=y
proof
  assume that
A1: congr a,b,c,x and
A2: congr a,b,c,y;
A3: now
    assume that
    a<>b and
A4: not a,b,c are_collinear;
    parallelogram a,b,c,x & parallelogram a,b,c,y by A1,A2,A4,Th59;
    hence thesis by Th45;
  end;
A5: now
    assume that
A6: a<>b and
A7: a,b,c are_collinear;
    consider p,q such that
A8: parallelogram a,b,p,q by A6,Lm1;
    parallelogram p,q,a,b by A8,Th43;
    then parallelogram p,q,c,x & parallelogram p,q,c,y by A1,A2,A7,Th61;
    hence thesis by Th45;
  end;
  now
    assume
A9: a=b;
    then c =x by A1,Th54;
    hence thesis by A2,A9,Th54;
  end;
  hence thesis by A5,A3;
end;
