reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;
reserve m,n for Nat,
  s,w for FinSequence of NAT;
reserve i,j,e,u for set,
        n for Nat;

theorem Th142:
  for a,b being set holds (a .--> b)*(n|->a) = n |-> b
proof
  let a,b be set;
A1: now
    let x be object;
    hereby
      assume x in dom (n |-> b);
      then
A2:   x in Seg n;
      hence x in dom(n|->a);
      dom(a .--> b) = {a} & (n|->a).x = a by A2,FUNCOP_1:7;
      hence (n|->a).x in dom(a .--> b) by TARSKI:def 1;
    end;
    assume that
A3: x in dom(n|->a) and
    (n|->a).x in dom(a .--> b);
    x in Seg n by A3;
    hence x in dom (n |-> b);
  end;
  now
    let x be object;
A4: a in {a} by TARSKI:def 1;
    assume x in dom (n |-> b);
    then
A5: x in Seg n;
    hence (n |-> b).x = b by FUNCOP_1:7
      .= (a .--> b).a by A4,FUNCOP_1:7
      .= (a .--> b).((n|->a).x) by A5,FUNCOP_1:7;
  end;
  hence thesis by A1,FUNCT_1:10;
end;
