theorem Th77:
  f is dilatation & f.a=a & f.b=b & not LIN a,b,x implies f.x=x
proof
  assume that
A1: f is dilatation and
A2: f.a=a and
A3: f.b=b and
A4: not LIN a,b,x;
  a,x // a,f.x by A1,A2,Th68;
  then LIN a,x,f.x by AFF_1:def 1;
  then
A5: LIN x,f.x,a by AFF_1:6;
  b,x // b,f.x by A1,A3,Th68;
  then LIN b,x,f.x by AFF_1:def 1;
  then
A6: LIN x,f.x,x & LIN x,f.x,b by AFF_1:6,7;
  assume f.x<>x;
  hence contradiction by A4,A5,A6,AFF_1:8;
end;
