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 Th74:
  f is dilatation & LIN x,f.x,y implies LIN x,f.x,f.y
proof
  assume
A1: f is dilatation;
  assume
A2: LIN x,f.x,y;
  now
    assume
A3: x<>y;
    x,f.x // x,y & x,y // f.x,f.y by A1,A2,Th68,AFF_1:def 1;
    then x,f.x // f.x,f.y by A3,AFF_1:5;
    then f.x,x // f.x,f.y by AFF_1:4;
    then LIN f.x,x,f.y by AFF_1:def 1;
    hence thesis by AFF_1:6;
  end;
  hence thesis by AFF_1:7;
end;
