reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th48:
  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 Th25;
  consider q such that
A2: parallelogram a,b,p,q by A1,Th44;
  not p,q,b are_collinear by A2,Th38;
  then consider c such that
A3: parallelogram p,q,b,c by Th44;
  take c;
A4: p,q // b,c by A3;
  p<>q & a,b // p,q by A2,Th36;
  then a,b // b,c by A4,Th8;
  then b,a // b,c by Th6;
  then
A5: b,a,c are_collinear;
A6: not a,q // b,p by A2,Th46;
  p,b // q,c & not b,c,p are_collinear by A3,Th37;
  hence thesis by A6,A5,Th6,Th22,Th24;
end;
