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;

theorem Th45:
  f is dilatation implies for a,b,c,d holds a,b '||' c,d iff f.a,f
  .b '||' f.c,f.d
proof
  assume
A1: f is dilatation;
  let a,b,c,d;
A2: c,d '||' f.c,f.d by A1,Th34;
A3: a,b '||' f.a,f.b by A1,Th34;
A4: now
A5: now
A6:   now
        assume f.c = f.d;
        then c = d by FUNCT_2:58;
        hence a,b '||' c,d by DIRAF:20;
      end;
      assume a,b '||' f.c,f.d;
      hence a,b '||' c,d by A2,A6,DIRAF:23;
    end;
A7: now
      assume f.a=f.b;
      then a=b by FUNCT_2:58;
      hence a,b '||' c,d by DIRAF:20;
    end;
    assume f.a,f.b '||' f.c,f.d;
    hence a,b '||' c,d by A3,A7,A5,DIRAF:23;
  end;
  now
A8: now
A9:   c = d implies f.a,f.b '||' f.c,f.d by DIRAF:20;
      assume f.a,f.b '||' c,d;
      hence f.a,f.b '||' f.c,f.d by A2,A9,DIRAF:23;
    end;
    assume a,b '||' c,d;
    then f.a,f.b '||' c,d or a=b by A3,DIRAF:23;
    hence f.a,f.b '||' f.c,f.d by A8,DIRAF:20;
  end;
  hence thesis by A4;
end;
