theorem Satz8p18Uniqueness:
  x is_foot a,b,c & y is_foot a,b,c implies x = y
  proof
    assume that
A1: x is_foot a,b,c and
A2: y is_foot a,b,c;
A3: Collinear a,b,y & are_orthogonal a,b,c,y by A2;
A4: c in Line(c,x) & c in Line(c,y) by GTARSKI3:83;
    are_orthogonal a,b,x,c,x & are_orthogonal a,b,y,c,y by A1,A3,Satz8p15;
    then are_orthogonal Line(a,b),x,Line(c,x) &
      are_orthogonal Line(a,b),y,Line(c,y);
    then right_angle c,x,y & right_angle c,y,x by A4,Satz8p2;
    hence thesis by Satz8p7;
  end;
