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 Th51:
  parallelogram a,b,c,d implies a,b congr c,d
proof
A1: a,b '||' a,b by PARSP_1:25;
  assume
A2: parallelogram a,b,c,d;
  then
A3: ( not a,c,b are_collinear)& a<>b by Th26,Th28;
  a<>c by A2,Th26;
  then consider p such that
A4: a,c,p are_collinear and
A5: a<>p and
A6: c <>p by Th38;
  a,c '||' a,p by A4;
  then consider q such that
A7: parallelogram a,p,b,q by A5,A1,A3,Th11,Th34;
  parallelogram a,b,p,q by A7,Th33;
  then parallelogram c,d,p,q by A2,A4,A6,Th41;
  then
A8: parallelogram p,q,c,d by Th33;
  parallelogram p,q,a,b by A7,Th33;
  hence thesis by A8;
end;
