reserve S,S9 for non void Signature,
  f,g for Function;

theorem
  for S1,S2,E being non empty Signature holds E is Extension of S1 & E
  is Extension of S2 iff S1 tolerates S2 & E is Extension of S1+*S2
proof
  let S1,S2,E be non empty Signature;
  set f1 = id the carrier of S1, g1 = id the carrier' of S1;
  set f2 = id the carrier of S2, g2 = id the carrier' of S2;
A1: E is Extension of S1 & E is Extension of S2 implies S1 tolerates S2
  proof
    assume that
A2: S1 is Subsignature of E and
A3: S2 is Subsignature of E;
A4: the Arity of S2 c= the Arity of E by A3,INSTALG1:11;
    the Arity of S1 c= the Arity of E by A2,INSTALG1:11;
    hence the Arity of S1 tolerates the Arity of S2 by A4,PARTFUN1:52;
A5: the ResultSort of S2 c= the ResultSort of E by A3,INSTALG1:11;
    the ResultSort of S1 c= the ResultSort of E by A2,INSTALG1:11;
    hence thesis by A5,PARTFUN1:52;
  end;
A6: the carrier of S1+*S2 = (the carrier of S1) \/ the carrier of S2 by
CIRCCOMB:def 2;
  the carrier of S2 c= (the carrier of S1) \/ the carrier of S2 by XBOOLE_1:7;
  then
A7: f2 c= id ((the carrier of S1) \/ the carrier of S2) by FUNCT_4:3;
A8: the carrier' of S1+*S2 = (the carrier' of S1) \/ the carrier' of S2 by
CIRCCOMB:def 2;
  the carrier of S1 c= (the carrier of S1) \/ the carrier of S2 by XBOOLE_1:7;
  then f1 c= id ((the carrier of S1) \/ the carrier of S2) by FUNCT_4:3;
  then
A9: f1 tolerates f2 by A7,PARTFUN1:52;
  E is Extension of S1 & E is Extension of S2 implies E is Extension of S1+*S2
  proof
    assume that
A10: f1, g1 form_morphism_between S1, E and
A11: f2, g2 form_morphism_between S2, E;
    f1+*f2, g1+*g2 form_morphism_between S1+*S2, E by A9,A10,A11,Th49;
    then id the carrier of S1+*S2, g1+*g2 form_morphism_between S1+*S2, E by A6
,FUNCT_4:22;
    hence id the carrier of S1+*S2, id the carrier' of S1+*S2
    form_morphism_between S1+*S2, E by A8,FUNCT_4:22;
  end;
  hence
  E is Extension of S1 & E is Extension of S2 implies S1 tolerates S2 & E
  is Extension of S1+*S2 by A1;
  assume S1 tolerates S2;
  then
A12: S1+*S2 is Extension of S1 by Th47;
  S1+*S2 is Extension of S2 by Th48;
  hence thesis by A12,Th46;
end;
