reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;
reserve O for connected non empty Poset;
reserve R,Q for array of O;
reserve T for non empty array of O;
reserve p,q,r,s for Element of dom T;

theorem Th70:
  p in q & q in s implies (T,p,q)incl.(p,s) = [q,s] & (T,p,q)incl.(q,s) = [p,s]
  proof assume
A1: p in q & q in s;
    set X = dom T;
    set i = id X;
    set f = Swap(i,p,q);
    set h = [:f,f:];
    set Y = (succ q)\p;
A2: dom i = X;
A3: s <> p & s <> q by A1;
    thus (T,p,q)incl.(p,s) = [f.p,f.s] by A1,Th65
    .= [f.p,i.s] by A3,Th33
    .= [f.p,s]
    .= [i.q,s] by A2,Th29
    .= [q,s];
    thus (T,p,q)incl.(q,s) = [f.q,f.s] by A1,Th65
    .= [f.q,i.s] by A3,Th33
    .= [f.q,s]
    .= [i.p,s] by A2,Th31
    .= [p,s];
  end;
