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 Th38:
  a<>b implies ex c st a,b,c are_collinear & c <>a & c <>b
proof
  assume a<>b;
  then consider p such that
A1: not a,b,p are_collinear by Th14;
  consider q such that
A2: parallelogram a,b,p,q by A1,Th34;
  not p,q,b are_collinear by A2,Th28;
  then consider c such that
A3: parallelogram p,q,b,c by Th34;
A4: p,q '||' b,c by A3;
  p<>q & a,b '||' p,q by A2,Th26;
  then a,b '||' b,c by A4,PARSP_1:26;
  then b,a '||' b,c by PARSP_1:23;
  then b,a,c are_collinear;
  then
A5: a,b,c are_collinear by Th10;
A6: not a,q '||' b,p by A2,Th36;
  p,b '||' q,c by A3;
  then
A7: a<>c by A6,PARSP_1:23;
  b<>c by A3,Th26;
  hence thesis by A7,A5;
end;
