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;
reserve A,C,K for Subset of AFS;

theorem Th91:
  x in f.:A iff ex y st y in A & f.y=x
proof
  thus x in f.:A implies ex y st y in A & f.y=x
  proof
    assume x in f.:A;
    then ex y being object st y in dom f & y in A & x=f.y by FUNCT_1:def 6;
    hence thesis;
  end;
  given y such that
A1: y in A & f.y=x;
  dom f = the carrier of AFS by FUNCT_2:52;
  hence thesis by A1,FUNCT_1:def 6;
end;
