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 Th55:
  f is translation & g is translation & f.a=g.a implies f=g
proof
  assume that
A1: f is translation and
A2: g is translation and
A3: f.a=g.a;
  set b=f.a;
A4: f is dilatation by A1;
A5: now
    assume
A6: a<>b;
    for x holds f.x=g.x
    proof
      let x;
      now
        assume
A7:     a,b,x are_collinear;
        now
A8:       f<>(id the carrier of OAS) by A6;
          then
A9:       f.x<>x by A1;
          assume x<>a;
          consider p such that
A10:      not a,b,p are_collinear by A6,DIRAF:37;
A11:      f.p<>p by A1,A8;
A12:      not p,f.p,x are_collinear
          proof
A13:        a,b,f.x are_collinear by A4,A7,Th47;
            a,b,a are_collinear by DIRAF:31;
            then
A14:        x,f.x,a are_collinear by A6,A7,A13,DIRAF:32;
A15:        p,f.p,p are_collinear by DIRAF:31;
            a,b,b are_collinear by DIRAF:31;
            then
A16:        x,f.x,b are_collinear by A6,A7,A13,DIRAF:32;
            assume
A17:        p,f.p,x are_collinear;
            then p,f.p,f.x are_collinear by A4,Th47;
            then x,f.x,p are_collinear by A11,A17,A15,DIRAF:32;
            hence contradiction by A10,A9,A14,A16,DIRAF:32;
          end;
          f.p=g.p by A1,A2,A3,A10,Th54;
          hence thesis by A1,A2,A12,Th54;
        end;
        hence thesis by A3;
      end;
      hence thesis by A1,A2,A3,Th54;
    end;
    hence thesis by FUNCT_2:63;
  end;
  b=a implies thesis by A1,A2,A3;
  hence thesis by A5;
end;
