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
  ex d st a,b congr c,d
proof
A1: now
    assume a=b;
    then a,b congr c,c;
    hence thesis;
  end;
A2: now
    assume that
A3: a<>b and
A4: a,b,c are_collinear;
    consider p,q such that
A5: parallelogram a,b,p,q by A3,Th43;
    not p,q,c are_collinear by A4,A5,Th29;
    then consider d such that
A6: parallelogram p,q,c,d by Th34;
    parallelogram p,q,a,b by A5,Th33;
    then a,b congr c,d by A6;
    hence thesis;
  end;
  now
    assume that
    a<>b and
A7: not a,b,c are_collinear;
    ex d st parallelogram a,b,c,d by A7,Th34;
    hence thesis by Th51;
  end;
  hence thesis by A1,A2;
end;
