reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;
reserve A for set,
  f,g,h for Function;

theorem
  A is functional &
  (for f,g being Function st f in A & g in A holds f tolerates g)
  implies union A is Function
proof
  assume that
A1: for x st x in A holds x is Function and
A2: for f,g being Function st f in A & g in A holds f tolerates g;
A3: now
    let x,y,z be object;
    assume that
A4: [x,y] in union A and
A5: [x,z] in union A;
    consider p being set such that
A6: [x,y] in p and
A7: p in A by A4,TARSKI:def 4;
    consider q being set such that
A8: [x,z] in q and
A9: q in A by A5,TARSKI:def 4;
    reconsider p,q as Function by A1,A7,A9;
    p tolerates q by A2,A7,A9;
    hence y = z by A6,A8,Th77;
  end;
  now
    let z be object;
    assume z in union A;
    then consider p being set such that
A10: z in p and
A11: p in A by TARSKI:def 4;
    reconsider p as Function by A1,A11;
    p = p;
    hence ex x,y being object st [x,y] = z by A10,RELAT_1:def 1;
  end;
  hence thesis by A3,FUNCT_1:def 1,RELAT_1:def 1;
end;
