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 Th84:
  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:     LIN a,b,x;
        now
A8:       f<>(id the carrier of AFS) by A6;
          then
A9:       f.x<>x by A1;
          assume x<>a;
          consider p such that
A10:      not LIN a,b,p by A6,AFF_1:13;
A11:      f.p<>p by A1,A8;
A12:      not LIN p,f.p,x
          proof
A13:        LIN a,b,f.x by A4,A7,Th74;
            LIN a,b,a by AFF_1:7;
            then
A14:        LIN x,f.x,a by A6,A7,A13,AFF_1:8;
A15:        LIN p,f.p,p by AFF_1:7;
            LIN a,b,b by AFF_1:7;
            then
A16:        LIN x,f.x,b by A6,A7,A13,AFF_1:8;
            assume
A17:        LIN p,f.p,x;
            then LIN p,f.p,f.x by A4,Th74;
            then LIN x,f.x,p by A11,A17,A15,AFF_1:8;
            hence contradiction by A10,A9,A14,A16,AFF_1:8;
          end;
          f.p=g.p by A1,A2,A3,A10,Th83;
          hence thesis by A1,A2,A12,Th83;
        end;
        hence thesis by A3;
      end;
      hence thesis by A1,A2,A3,Th83;
    end;
    hence thesis by FUNCT_2:63;
  end;
  b=a implies thesis by A1,A2,A3;
  hence thesis by A5;
end;
