reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;
reserve CS for CongrSpace;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;
reserve AFS for AffinSpace;
reserve a,b,c,d,d1,d2,p,x,y,z,t for Element of AFS;
reserve f,g for Permutation of the carrier of AFS;

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;
