reserve F for Field;
reserve a,b,c,d,p,q,r for Element of MPS(F);
reserve e,f,g,h,i,j,k,l,m,n,o,w for Element of [:the carrier of F,the carrier
  of F,the carrier of F:];
reserve K,L,M,N,R,S for Element of F;
reserve FdSp for FanodesSp;
reserve a,b,c,d,p,q,r,s,o,x,y for Element of FdSp;

theorem
  a,b congr c,x & a,b congr c,y implies x=y
proof
  assume that
A1: a,b congr c,x and
A2: a,b congr 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,Th50;
    hence thesis by Th35;
  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,Th43;
    parallelogram p,q,a,b by A8,Th33;
    then parallelogram p,q,c,x & parallelogram p,q,c,y by A1,A2,A7,Th52;
    hence thesis by Th35;
  end;
  now
    assume
A9: a=b;
    then c =x by A1,Th45;
    hence thesis by A2,A9,Th45;
  end;
  hence thesis by A5,A3;
end;
