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 Th76:
  f is dilatation implies ((f=id the carrier of AFS or for x holds
  f.x<>x) iff for x,y holds x,f.x // y,f.y )
proof
  assume
A1: f is dilatation;
A2: now
    assume
A3: for x,y holds x,f.x // y,f.y;
    assume f<>(id the carrier of AFS);
    then consider y such that
A4: f.y<>(id the carrier of AFS).y by FUNCT_2:63;
    given x such that
A5: f.x=x;
    x<>y by A5,A4;
    then consider z such that
A6: not LIN x,y,z by AFF_1:13;
    x,z // f.x,f.z by A1,Th68;
    then LIN x,z,f.z by A5,AFF_1:def 1;
    then
A7: LIN z,f.z,x by AFF_1:6;
    LIN z,f.z,z by AFF_1:7;
    then
A8: z,f.z // x,z by A7,AFF_1:10;
A9: f.y<>y by A4;
    x,y // f.x,f.y by A1,Th68;
    then
A10: LIN x,y,f.y by A5,AFF_1:def 1;
    then LIN y,x,f.y by AFF_1:6;
    then
A11: y,x // y,f.y by AFF_1:def 1;
A12: LIN y,f.y,x by A10,AFF_1:6;
A13: now
      assume z=f.z;
      then z,y //z,f.y by A1,Th68;
      then LIN z,y,f.y by AFF_1:def 1;
      then LIN y,f.y,y & LIN y,f.y,z by AFF_1:6,7;
      hence contradiction by A9,A12,A6,AFF_1:8;
    end;
    y,f.y // z,f.z by A3;
    then y,f.y // x,z by A13,A8,AFF_1:5;
    then y,x // x,z by A9,A11,AFF_1:5;
    then x,y // x,z by AFF_1:4;
    hence contradiction by A6,AFF_1:def 1;
  end;
  now
    assume
A14: f=id the carrier of AFS or for z holds f.z<>z;
    let x,y;
A15: x,y // f.x,f.y by A1,Th68;
A16: now
      assume
A17:  for z holds f.z<>z;
      assume
A18:  not x,f.x // y,f.y;
      then consider z such that
A19:  LIN x,f.x,z and
A20:  LIN y,f.y,z by A15,Th75;
      set t=f.z;
      LIN x,f.x,t by A1,A19,Th74;
      then
A21:  x,f.x // z,t by A19,AFF_1:10;
      LIN y,f.y,t by A1,A20,Th74;
      then
A22:  y,f.y // z,t by A20,AFF_1:10;
      z<>t by A17;
      hence contradiction by A18,A21,A22,AFF_1:5;
    end;
    f=(id the carrier of AFS) implies x,f.x // y,f.y by AFF_1:3;
    hence x,f.x // y,f.y by A14,A16;
  end;
  hence thesis by A2;
end;
