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;

theorem Th47:
  inversions R c= [:dom R, dom R:]
  proof
    let x be object; assume
A1: x in inversions R; then
    consider a,b being Element of dom R such that
A2: x = [a,b] & a in b & R/.a > R/.b;
    a in dom R & b in dom R by A2,A1,Th46;
    hence thesis by A2,ZFMISC_1:def 2;
  end;
