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
  r,s congr a,b & r,s congr c,d implies a,b congr c,d
proof
  assume that
A1: r,s congr a,b and
A2: r,s congr 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,Th50;
    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,Th52;
    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,Th52;
    hence thesis by A14;
  end;
  r=s implies thesis by A1,A2,Th45;
  hence thesis by A10,A5,A3;
end;
