reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;

theorem Th20: f={[x,f.x] where x is Element of dom f: x in dom f}
proof
set RH={[x,f.x] where x is Element of dom f: x in dom f};
now let z be object; assume
A1: z in f; then consider x, y being object such that
A2: z=[x,y] by RELAT_1:def 1;
reconsider xx=x as Element of dom f by FUNCT_1:1, A2, A1;
z=[xx,f.xx] & xx in dom f by A2, FUNCT_1:1, A1;
hence z in RH;
end; then
A3: f c= RH;
now let z be object; assume
A4: z in RH;
consider x being Element of dom f such that
A5: z=[x,f.x] & x in dom f by A4;
thus z in f by A5, FUNCT_1:1;
end;
then RH c= f; hence thesis by  A3;
end;
