 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem Th77:
  x is Element of GPerms X iff x is Permutation of X
proof
A1: x is Permutation of X implies x in Funcs(X,X) by FUNCT_2:9;
  carr(GPerms X) c= carr(GFuncs X) by Th23;
  then
A2: x is Element of GPerms X implies x is Element of GFuncs X;
  carr(GFuncs X) = Funcs(X,X) by Def40;
  hence thesis by A1,A2,Def42;
end;
