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 for Q being quasi_total Relation of B, U1, ::#Th27
R being quasi_total Relation of B, U2, P being Relation of
A, B st P*Q*(Q~)*R is Function-like holds P*Q*(Q~)*R=P*R
proof
let Q be quasi_total Relation of B, U1;
let  R be quasi_total Relation of B, U2; let P be Relation of A, B;
reconsider QQ=Q as total B-defined Relation;
reconsider RR=R as total B-defined Relation;
A1: dom R=B & rng P c= B by PARTFUN1:def 2;
assume P*Q*(Q~)*R is Function-like; hence thesis by A1, Lm43;
end;
