reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

theorem Th20:
  (i in dom p implies [#](p,d).i = p.i) & (not i in dom p implies
  [#](p,d).i = d)
proof
  thus i in dom p implies [#](p,d).i = p.i by FUNCT_4:13;
A1: i in NAT by ORDINAL1:def 12;
  assume not i in dom p;
  hence [#](p,d).i = (NAT --> d).i by FUNCT_4:11
    .= d by A1,FUNCOP_1:7;
end;
