reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem Th26:
  f c= g implies min(f,g) = f
proof
  assume
A1: f c= g;
A2: for x being Element of C st x in C holds min(f,g).x = f.x
  proof
    let x be Element of C;
    f.x <= g.x by A1;
    then f.x = min(f.x,g.x) by XXREAL_0:def 9
      .= min(f,g).x by Def3;
    hence thesis;
  end;
  C = dom min(f,g) & C = dom f by FUNCT_2:def 1;
  hence thesis by A2,PARTFUN1:5;
end;
