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 Th85:
  f is translation implies f" is translation
proof
A1: now
    assume
A2: for x holds f.x<>x;
    given y such that
A3: f".y=y;
    f.y=y by A3,Th2;
    hence contradiction by A2;
  end;
  assume
A4: f is translation;
  then f is dilatation;
  then
A5: f" is dilatation by Th70;
  f=(id the carrier of AFS) implies f"=(id the carrier of AFS) by FUNCT_1:45;
  hence thesis by A4,A5,A1;
end;
