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 Th48:
  a,b congr c,d implies a,b '||' c,d
proof
  assume
A1: a,b congr c,d;
  now
    assume a<>b;
    then consider p,q such that
A2: parallelogram p,q,a,b and
A3: parallelogram p,q,c,d by A1;
A4: p,q '||' c,d by A3;
    p<>q & p,q '||' a,b by A2,Th26;
    hence thesis by A4,PARSP_1:def 12;
  end;
  hence thesis by PARSP_1:20;
end;
