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 Th52:
  a,b congr c,d & a,b,c are_collinear & parallelogram r,s,a,b
  implies parallelogram r,s,c,d
proof
  assume that
A1: a,b congr c,d and
A2: a,b,c are_collinear and
A3: parallelogram r,s,a,b;
  now
    assume
A4: a<>b;
    then consider p,q such that
A5: parallelogram p,q,a,b and
A6: parallelogram p,q,c,d by A1;
    r,s '||' a,b & a,b '||' c,d by A1,A3,Th48;
    then
A7: r,s '||' c,d by A4,PARSP_1:26;
A8: parallelogram a,b,r,s by A3,Th33;
    parallelogram a,b,p,q by A5,Th33;
    then
A9: r,c '||' s,d by A6,A8,Th42;
    not r,s,c are_collinear by A2,A3,Th29;
    hence thesis by A7,A9;
  end;
  hence thesis by A3,Th26;
end;
